(atan (* (/ z1 z2) (tan (* PI (+ 1/2 (+ z0 z0))))))

Percentage Accurate: 32.3% → 93.9%

Time: 7.0s

Alternatives: 23

Speedup: 1.7×

Specification

Math

\[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (atan (* (/ z1 z2) (tan (* PI (+ 0.5 (+ z0 z0)))))))

C

double code(double z1, double z2, double z0) {
	return atan(((z1 / z2) * tan((((double) M_PI) * (0.5 + (z0 + z0))))));
}

Java

public static double code(double z1, double z2, double z0) {
	return Math.atan(((z1 / z2) * Math.tan((Math.PI * (0.5 + (z0 + z0))))));
}

Python

def code(z1, z2, z0):
	return math.atan(((z1 / z2) * math.tan((math.pi * (0.5 + (z0 + z0))))))

Julia

function code(z1, z2, z0)
	return atan(Float64(Float64(z1 / z2) * tan(Float64(pi * Float64(0.5 + Float64(z0 + z0))))))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = atan(((z1 / z2) * tan((pi * (0.5 + (z0 + z0))))));
end

Wolfram

code[z1_, z2_, z0_] := N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(Pi * N[(0.5 + N[(z0 + z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 32.3% accurate, 1.0× speedup

Math

\[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (atan (* (/ z1 z2) (tan (* PI (+ 0.5 (+ z0 z0)))))))

C

double code(double z1, double z2, double z0) {
	return atan(((z1 / z2) * tan((((double) M_PI) * (0.5 + (z0 + z0))))));
}

Java

public static double code(double z1, double z2, double z0) {
	return Math.atan(((z1 / z2) * Math.tan((Math.PI * (0.5 + (z0 + z0))))));
}

Python

def code(z1, z2, z0):
	return math.atan(((z1 / z2) * math.tan((math.pi * (0.5 + (z0 + z0))))))

Julia

function code(z1, z2, z0)
	return atan(Float64(Float64(z1 / z2) * tan(Float64(pi * Float64(0.5 + Float64(z0 + z0))))))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = atan(((z1 / z2) * tan((pi * (0.5 + (z0 + z0))))));
end

Wolfram

code[z1_, z2_, z0_] := N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(Pi * N[(0.5 + N[(z0 + z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \pi\right)\\ t_1 := {t\_0}^{2}\\ t_2 := \sin \left(0.5 \cdot \pi\right)\\ t_3 := {t\_2}^{2}\\ t_4 := -2 \cdot \frac{\pi \cdot t\_3}{t\_1}\\ t_5 := 2 \cdot \pi - t\_4\\ t_6 := \pi \cdot \left(t\_2 \cdot t\_5\right)\\ t_7 := z2 \cdot t\_0\\ \mathbf{if}\;z0 \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1}{z2} \cdot \left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot t\_6}{t\_0}\right) - t\_4\right) + \frac{t\_2}{t\_0}\right)\right)\\ \mathbf{elif}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(z0 \cdot \left(z0 \cdot \left(2 \cdot \frac{z1 \cdot t\_6}{t\_7} + \frac{z0 \cdot \left(z1 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left(t\_3 \cdot t\_5\right)}{t\_1} + \left(-2 \cdot \left({\pi}^{2} \cdot t\_5\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot t\_3}{t\_1}\right)\right)\right)\right)}{z2}\right) + \frac{z1 \cdot t\_5}{z2}\right) + \frac{z1 \cdot t\_2}{t\_7}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (cos (* 0.5 PI)))
       (t_1 (pow t_0 2.0))
       (t_2 (sin (* 0.5 PI)))
       (t_3 (pow t_2 2.0))
       (t_4 (* -2.0 (/ (* PI t_3) t_1)))
       (t_5 (- (* 2.0 PI) t_4))
       (t_6 (* PI (* t_2 t_5)))
       (t_7 (* z2 t_0)))
  (if (<= z0 -9e+18)
    (atan
     (*
      (/ z1 z2)
      (+
       (* z0 (- (+ (* 2.0 PI) (* 2.0 (/ (* z0 t_6) t_0))) t_4))
       (/ t_2 t_0))))
    (if (<= z0 1.5e+14)
      (atan
       (*
        (- z1)
        (/
         (sin (+ (* (* PI -2.0) z0) (* 0.5 PI)))
         (* (- z2) (sin (* (* -2.0 z0) PI))))))
      (atan
       (+
        (*
         z0
         (+
          (*
           z0
           (+
            (* 2.0 (/ (* z1 t_6) t_7))
            (/
             (*
              z0
              (*
               z1
               (-
                (* -1.3333333333333333 (pow PI 3.0))
                (+
                 (* -4.0 (/ (* (pow PI 2.0) (* t_3 t_5)) t_1))
                 (+
                  (* -2.0 (* (pow PI 2.0) t_5))
                  (*
                   1.3333333333333333
                   (/ (* (pow PI 3.0) t_3) t_1)))))))
             z2)))
          (/ (* z1 t_5) z2)))
        (/ (* z1 t_2) t_7)))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = cos((0.5 * ((double) M_PI)));
	double t_1 = pow(t_0, 2.0);
	double t_2 = sin((0.5 * ((double) M_PI)));
	double t_3 = pow(t_2, 2.0);
	double t_4 = -2.0 * ((((double) M_PI) * t_3) / t_1);
	double t_5 = (2.0 * ((double) M_PI)) - t_4;
	double t_6 = ((double) M_PI) * (t_2 * t_5);
	double t_7 = z2 * t_0;
	double tmp;
	if (z0 <= -9e+18) {
		tmp = atan(((z1 / z2) * ((z0 * (((2.0 * ((double) M_PI)) + (2.0 * ((z0 * t_6) / t_0))) - t_4)) + (t_2 / t_0))));
	} else if (z0 <= 1.5e+14) {
		tmp = atan((-z1 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / (-z2 * sin(((-2.0 * z0) * ((double) M_PI)))))));
	} else {
		tmp = atan(((z0 * ((z0 * ((2.0 * ((z1 * t_6) / t_7)) + ((z0 * (z1 * ((-1.3333333333333333 * pow(((double) M_PI), 3.0)) - ((-4.0 * ((pow(((double) M_PI), 2.0) * (t_3 * t_5)) / t_1)) + ((-2.0 * (pow(((double) M_PI), 2.0) * t_5)) + (1.3333333333333333 * ((pow(((double) M_PI), 3.0) * t_3) / t_1))))))) / z2))) + ((z1 * t_5) / z2))) + ((z1 * t_2) / t_7)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = Math.cos((0.5 * Math.PI));
	double t_1 = Math.pow(t_0, 2.0);
	double t_2 = Math.sin((0.5 * Math.PI));
	double t_3 = Math.pow(t_2, 2.0);
	double t_4 = -2.0 * ((Math.PI * t_3) / t_1);
	double t_5 = (2.0 * Math.PI) - t_4;
	double t_6 = Math.PI * (t_2 * t_5);
	double t_7 = z2 * t_0;
	double tmp;
	if (z0 <= -9e+18) {
		tmp = Math.atan(((z1 / z2) * ((z0 * (((2.0 * Math.PI) + (2.0 * ((z0 * t_6) / t_0))) - t_4)) + (t_2 / t_0))));
	} else if (z0 <= 1.5e+14) {
		tmp = Math.atan((-z1 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / (-z2 * Math.sin(((-2.0 * z0) * Math.PI))))));
	} else {
		tmp = Math.atan(((z0 * ((z0 * ((2.0 * ((z1 * t_6) / t_7)) + ((z0 * (z1 * ((-1.3333333333333333 * Math.pow(Math.PI, 3.0)) - ((-4.0 * ((Math.pow(Math.PI, 2.0) * (t_3 * t_5)) / t_1)) + ((-2.0 * (Math.pow(Math.PI, 2.0) * t_5)) + (1.3333333333333333 * ((Math.pow(Math.PI, 3.0) * t_3) / t_1))))))) / z2))) + ((z1 * t_5) / z2))) + ((z1 * t_2) / t_7)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = math.cos((0.5 * math.pi))
	t_1 = math.pow(t_0, 2.0)
	t_2 = math.sin((0.5 * math.pi))
	t_3 = math.pow(t_2, 2.0)
	t_4 = -2.0 * ((math.pi * t_3) / t_1)
	t_5 = (2.0 * math.pi) - t_4
	t_6 = math.pi * (t_2 * t_5)
	t_7 = z2 * t_0
	tmp = 0
	if z0 <= -9e+18:
		tmp = math.atan(((z1 / z2) * ((z0 * (((2.0 * math.pi) + (2.0 * ((z0 * t_6) / t_0))) - t_4)) + (t_2 / t_0))))
	elif z0 <= 1.5e+14:
		tmp = math.atan((-z1 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / (-z2 * math.sin(((-2.0 * z0) * math.pi))))))
	else:
		tmp = math.atan(((z0 * ((z0 * ((2.0 * ((z1 * t_6) / t_7)) + ((z0 * (z1 * ((-1.3333333333333333 * math.pow(math.pi, 3.0)) - ((-4.0 * ((math.pow(math.pi, 2.0) * (t_3 * t_5)) / t_1)) + ((-2.0 * (math.pow(math.pi, 2.0) * t_5)) + (1.3333333333333333 * ((math.pow(math.pi, 3.0) * t_3) / t_1))))))) / z2))) + ((z1 * t_5) / z2))) + ((z1 * t_2) / t_7)))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = cos(Float64(0.5 * pi))
	t_1 = t_0 ^ 2.0
	t_2 = sin(Float64(0.5 * pi))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(-2.0 * Float64(Float64(pi * t_3) / t_1))
	t_5 = Float64(Float64(2.0 * pi) - t_4)
	t_6 = Float64(pi * Float64(t_2 * t_5))
	t_7 = Float64(z2 * t_0)
	tmp = 0.0
	if (z0 <= -9e+18)
		tmp = atan(Float64(Float64(z1 / z2) * Float64(Float64(z0 * Float64(Float64(Float64(2.0 * pi) + Float64(2.0 * Float64(Float64(z0 * t_6) / t_0))) - t_4)) + Float64(t_2 / t_0))));
	elseif (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(-z1) * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi))))));
	else
		tmp = atan(Float64(Float64(z0 * Float64(Float64(z0 * Float64(Float64(2.0 * Float64(Float64(z1 * t_6) / t_7)) + Float64(Float64(z0 * Float64(z1 * Float64(Float64(-1.3333333333333333 * (pi ^ 3.0)) - Float64(Float64(-4.0 * Float64(Float64((pi ^ 2.0) * Float64(t_3 * t_5)) / t_1)) + Float64(Float64(-2.0 * Float64((pi ^ 2.0) * t_5)) + Float64(1.3333333333333333 * Float64(Float64((pi ^ 3.0) * t_3) / t_1))))))) / z2))) + Float64(Float64(z1 * t_5) / z2))) + Float64(Float64(z1 * t_2) / t_7)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = cos((0.5 * pi));
	t_1 = t_0 ^ 2.0;
	t_2 = sin((0.5 * pi));
	t_3 = t_2 ^ 2.0;
	t_4 = -2.0 * ((pi * t_3) / t_1);
	t_5 = (2.0 * pi) - t_4;
	t_6 = pi * (t_2 * t_5);
	t_7 = z2 * t_0;
	tmp = 0.0;
	if (z0 <= -9e+18)
		tmp = atan(((z1 / z2) * ((z0 * (((2.0 * pi) + (2.0 * ((z0 * t_6) / t_0))) - t_4)) + (t_2 / t_0))));
	elseif (z0 <= 1.5e+14)
		tmp = atan((-z1 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / (-z2 * sin(((-2.0 * z0) * pi))))));
	else
		tmp = atan(((z0 * ((z0 * ((2.0 * ((z1 * t_6) / t_7)) + ((z0 * (z1 * ((-1.3333333333333333 * (pi ^ 3.0)) - ((-4.0 * (((pi ^ 2.0) * (t_3 * t_5)) / t_1)) + ((-2.0 * ((pi ^ 2.0) * t_5)) + (1.3333333333333333 * (((pi ^ 3.0) * t_3) / t_1))))))) / z2))) + ((z1 * t_5) / z2))) + ((z1 * t_2) / t_7)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * N[(N[(Pi * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * Pi), $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(Pi * N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(z2 * t$95$0), $MachinePrecision]}, If[LessEqual[z0, -9e+18], N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[(N[(z0 * N[(N[(N[(2.0 * Pi), $MachinePrecision] + N[(2.0 * N[(N[(z0 * t$95$6), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.5e+14], N[ArcTan[N[((-z1) * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(z0 * N[(N[(z0 * N[(N[(2.0 * N[(N[(z1 * t$95$6), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 * N[(z1 * N[(N[(-1.3333333333333333 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(1.3333333333333333 * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z1 * t$95$5), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z1 * t$95$2), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -9e18

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \color{blue}{\left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot \left(\pi \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right) - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right)}\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \color{blue}{\left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot \left(\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)}\right) \]

   if -9e18 < z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(2 \cdot \frac{z1 \cdot \left(\pi \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)\right)}{z2 \cdot \cos \left(\frac{1}{2} \cdot \pi\right)} + \frac{z0 \cdot \left(z1 \cdot \left(\frac{-4}{3} \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(\frac{1}{2} \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right) + \frac{4}{3} \cdot \frac{{\pi}^{3} \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{z2}\right) + \frac{z1 \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)}{z2}\right) + \frac{z1 \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{z2 \cdot \cos \left(\frac{1}{2} \cdot \pi\right)}\right)} \]

   4. Applied rewrites 33.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(2 \cdot \frac{z1 \cdot \left(\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)}{z2 \cdot \cos \left(0.5 \cdot \pi\right)} + \frac{z0 \cdot \left(z1 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{z2}\right) + \frac{z1 \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)}{z2}\right) + \frac{z1 \cdot \sin \left(0.5 \cdot \pi\right)}{z2 \cdot \cos \left(0.5 \cdot \pi\right)}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{{\pi}^{3}}{z2}\\ t_1 := \cos \left(0.5 \cdot \pi\right)\\ t_2 := \sin \left(0.5 \cdot \pi\right)\\ t_3 := \frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\\ t_4 := -2 \cdot \frac{\pi \cdot {t\_2}^{2}}{{t\_1}^{2}}\\ \mathbf{if}\;z0 \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1}{z2} \cdot \left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot \left(\pi \cdot \left(t\_2 \cdot \left(2 \cdot \pi - t\_4\right)\right)\right)}{t\_1}\right) - t\_4\right) + \frac{t\_2}{t\_1}\right)\right)\\ \mathbf{elif}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(-0.3333333333333333 \cdot t\_0 - \left(-0.6666666666666666 \cdot \left({\pi}^{2} \cdot t\_3\right) + -0.06666666666666667 \cdot t\_0\right)\right)\right) + -1 \cdot t\_3\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (/ (pow PI 3.0) z2))
       (t_1 (cos (* 0.5 PI)))
       (t_2 (sin (* 0.5 PI)))
       (t_3 (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
       (t_4 (* -2.0 (/ (* PI (pow t_2 2.0)) (pow t_1 2.0)))))
  (if (<= z0 -9e+18)
    (atan
     (*
      (/ z1 z2)
      (+
       (*
        z0
        (-
         (+
          (* 2.0 PI)
          (* 2.0 (/ (* z0 (* PI (* t_2 (- (* 2.0 PI) t_4)))) t_1)))
         t_4))
       (/ t_2 t_1))))
    (if (<= z0 1.5e+14)
      (atan
       (*
        (- z1)
        (/
         (sin (+ (* (* PI -2.0) z0) (* 0.5 PI)))
         (* (- z2) (sin (* (* -2.0 z0) PI))))))
      (atan
       (*
        (- z1)
        (/
         (+
          (*
           (pow z0 2.0)
           (+
            (*
             -1.0
             (*
              (pow z0 2.0)
              (-
               (* -0.3333333333333333 t_0)
               (+
                (* -0.6666666666666666 (* (pow PI 2.0) t_3))
                (* -0.06666666666666667 t_0)))))
            (* -1.0 t_3)))
          (* 0.5 (/ 1.0 (* z2 PI))))
         z0)))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = pow(((double) M_PI), 3.0) / z2;
	double t_1 = cos((0.5 * ((double) M_PI)));
	double t_2 = sin((0.5 * ((double) M_PI)));
	double t_3 = (((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2));
	double t_4 = -2.0 * ((((double) M_PI) * pow(t_2, 2.0)) / pow(t_1, 2.0));
	double tmp;
	if (z0 <= -9e+18) {
		tmp = atan(((z1 / z2) * ((z0 * (((2.0 * ((double) M_PI)) + (2.0 * ((z0 * (((double) M_PI) * (t_2 * ((2.0 * ((double) M_PI)) - t_4)))) / t_1))) - t_4)) + (t_2 / t_1))));
	} else if (z0 <= 1.5e+14) {
		tmp = atan((-z1 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / (-z2 * sin(((-2.0 * z0) * ((double) M_PI)))))));
	} else {
		tmp = atan((-z1 * (((pow(z0, 2.0) * ((-1.0 * (pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (pow(((double) M_PI), 2.0) * t_3)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_3))) + (0.5 * (1.0 / (z2 * ((double) M_PI))))) / z0)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = Math.pow(Math.PI, 3.0) / z2;
	double t_1 = Math.cos((0.5 * Math.PI));
	double t_2 = Math.sin((0.5 * Math.PI));
	double t_3 = (Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2));
	double t_4 = -2.0 * ((Math.PI * Math.pow(t_2, 2.0)) / Math.pow(t_1, 2.0));
	double tmp;
	if (z0 <= -9e+18) {
		tmp = Math.atan(((z1 / z2) * ((z0 * (((2.0 * Math.PI) + (2.0 * ((z0 * (Math.PI * (t_2 * ((2.0 * Math.PI) - t_4)))) / t_1))) - t_4)) + (t_2 / t_1))));
	} else if (z0 <= 1.5e+14) {
		tmp = Math.atan((-z1 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / (-z2 * Math.sin(((-2.0 * z0) * Math.PI))))));
	} else {
		tmp = Math.atan((-z1 * (((Math.pow(z0, 2.0) * ((-1.0 * (Math.pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (Math.pow(Math.PI, 2.0) * t_3)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_3))) + (0.5 * (1.0 / (z2 * Math.PI)))) / z0)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = math.pow(math.pi, 3.0) / z2
	t_1 = math.cos((0.5 * math.pi))
	t_2 = math.sin((0.5 * math.pi))
	t_3 = (math.pi / z2) - (0.3333333333333333 * (math.pi / z2))
	t_4 = -2.0 * ((math.pi * math.pow(t_2, 2.0)) / math.pow(t_1, 2.0))
	tmp = 0
	if z0 <= -9e+18:
		tmp = math.atan(((z1 / z2) * ((z0 * (((2.0 * math.pi) + (2.0 * ((z0 * (math.pi * (t_2 * ((2.0 * math.pi) - t_4)))) / t_1))) - t_4)) + (t_2 / t_1))))
	elif z0 <= 1.5e+14:
		tmp = math.atan((-z1 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / (-z2 * math.sin(((-2.0 * z0) * math.pi))))))
	else:
		tmp = math.atan((-z1 * (((math.pow(z0, 2.0) * ((-1.0 * (math.pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (math.pow(math.pi, 2.0) * t_3)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_3))) + (0.5 * (1.0 / (z2 * math.pi)))) / z0)))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64((pi ^ 3.0) / z2)
	t_1 = cos(Float64(0.5 * pi))
	t_2 = sin(Float64(0.5 * pi))
	t_3 = Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))
	t_4 = Float64(-2.0 * Float64(Float64(pi * (t_2 ^ 2.0)) / (t_1 ^ 2.0)))
	tmp = 0.0
	if (z0 <= -9e+18)
		tmp = atan(Float64(Float64(z1 / z2) * Float64(Float64(z0 * Float64(Float64(Float64(2.0 * pi) + Float64(2.0 * Float64(Float64(z0 * Float64(pi * Float64(t_2 * Float64(Float64(2.0 * pi) - t_4)))) / t_1))) - t_4)) + Float64(t_2 / t_1))));
	elseif (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(-z1) * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi))))));
	else
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64((z0 ^ 2.0) * Float64(Float64(-1.0 * Float64((z0 ^ 2.0) * Float64(Float64(-0.3333333333333333 * t_0) - Float64(Float64(-0.6666666666666666 * Float64((pi ^ 2.0) * t_3)) + Float64(-0.06666666666666667 * t_0))))) + Float64(-1.0 * t_3))) + Float64(0.5 * Float64(1.0 / Float64(z2 * pi)))) / z0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (pi ^ 3.0) / z2;
	t_1 = cos((0.5 * pi));
	t_2 = sin((0.5 * pi));
	t_3 = (pi / z2) - (0.3333333333333333 * (pi / z2));
	t_4 = -2.0 * ((pi * (t_2 ^ 2.0)) / (t_1 ^ 2.0));
	tmp = 0.0;
	if (z0 <= -9e+18)
		tmp = atan(((z1 / z2) * ((z0 * (((2.0 * pi) + (2.0 * ((z0 * (pi * (t_2 * ((2.0 * pi) - t_4)))) / t_1))) - t_4)) + (t_2 / t_1))));
	elseif (z0 <= 1.5e+14)
		tmp = atan((-z1 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / (-z2 * sin(((-2.0 * z0) * pi))))));
	else
		tmp = atan((-z1 * ((((z0 ^ 2.0) * ((-1.0 * ((z0 ^ 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * ((pi ^ 2.0) * t_3)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_3))) + (0.5 * (1.0 / (z2 * pi)))) / z0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] / z2), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * N[(N[(Pi * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -9e+18], N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[(N[(z0 * N[(N[(N[(2.0 * Pi), $MachinePrecision] + N[(2.0 * N[(N[(z0 * N[(Pi * N[(t$95$2 * N[(N[(2.0 * Pi), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.5e+14], N[ArcTan[N[((-z1) * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-z1) * N[(N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-1.0 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * t$95$0), $MachinePrecision] - N[(N[(-0.6666666666666666 * N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(-0.06666666666666667 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -9e18

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \color{blue}{\left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot \left(\pi \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right) - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right)}\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \color{blue}{\left(z0 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z0 \cdot \left(\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)}\right) \]

   if -9e18 < z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\pi}^{3}}{z2} - \left(\frac{-2}{3} \cdot \left({\pi}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right)\right) + \frac{-1}{15} \cdot \frac{{\pi}^{3}}{z2}\right)\right)\right) + -1 \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right)\right) + \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(-0.3333333333333333 \cdot \frac{{\pi}^{3}}{z2} - \left(-0.6666666666666666 \cdot \left({\pi}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + -0.06666666666666667 \cdot \frac{{\pi}^{3}}{z2}\right)\right)\right) + -1 \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{{\pi}^{3}}{z2}\\ t_1 := \left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\\ t_2 := \frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\\ \mathbf{if}\;z0 \leq -1.35 \cdot 10^{+247}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}{t\_1}\right)\\ \mathbf{elif}\;z0 \leq -21000000000:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{t\_1}\right)\\ \mathbf{elif}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(-0.3333333333333333 \cdot t\_0 - \left(-0.6666666666666666 \cdot \left({\pi}^{2} \cdot t\_2\right) + -0.06666666666666667 \cdot t\_0\right)\right)\right) + -1 \cdot t\_2\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (/ (pow PI 3.0) z2))
       (t_1 (* (- z2) (sin (* (* -2.0 z0) PI))))
       (t_2 (- (/ PI z2) (* 0.3333333333333333 (/ PI z2)))))
  (if (<= z0 -1.35e+247)
    (atan
     (*
      (- z1)
      (/
       (+
        1.0
        (*
         (pow z0 2.0)
         (+
          (* -2.0 (pow PI 2.0))
          (* 0.6666666666666666 (* (pow z0 2.0) (pow PI 4.0))))))
       t_1)))
    (if (<= z0 -21000000000.0)
      (atan
       (*
        (- z1)
        (/
         (+ 1.0 (* -2.0 (log (pow 19333.689074365146 (* z0 z0)))))
         t_1)))
      (if (<= z0 1.5e+14)
        (atan
         (* (- z1) (/ (sin (+ (* (* PI -2.0) z0) (* 0.5 PI))) t_1)))
        (atan
         (*
          (- z1)
          (/
           (+
            (*
             (pow z0 2.0)
             (+
              (*
               -1.0
               (*
                (pow z0 2.0)
                (-
                 (* -0.3333333333333333 t_0)
                 (+
                  (* -0.6666666666666666 (* (pow PI 2.0) t_2))
                  (* -0.06666666666666667 t_0)))))
              (* -1.0 t_2)))
            (* 0.5 (/ 1.0 (* z2 PI))))
           z0))))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = pow(((double) M_PI), 3.0) / z2;
	double t_1 = -z2 * sin(((-2.0 * z0) * ((double) M_PI)));
	double t_2 = (((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2));
	double tmp;
	if (z0 <= -1.35e+247) {
		tmp = atan((-z1 * ((1.0 + (pow(z0, 2.0) * ((-2.0 * pow(((double) M_PI), 2.0)) + (0.6666666666666666 * (pow(z0, 2.0) * pow(((double) M_PI), 4.0)))))) / t_1)));
	} else if (z0 <= -21000000000.0) {
		tmp = atan((-z1 * ((1.0 + (-2.0 * log(pow(19333.689074365146, (z0 * z0))))) / t_1)));
	} else if (z0 <= 1.5e+14) {
		tmp = atan((-z1 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / t_1)));
	} else {
		tmp = atan((-z1 * (((pow(z0, 2.0) * ((-1.0 * (pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (pow(((double) M_PI), 2.0) * t_2)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_2))) + (0.5 * (1.0 / (z2 * ((double) M_PI))))) / z0)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = Math.pow(Math.PI, 3.0) / z2;
	double t_1 = -z2 * Math.sin(((-2.0 * z0) * Math.PI));
	double t_2 = (Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2));
	double tmp;
	if (z0 <= -1.35e+247) {
		tmp = Math.atan((-z1 * ((1.0 + (Math.pow(z0, 2.0) * ((-2.0 * Math.pow(Math.PI, 2.0)) + (0.6666666666666666 * (Math.pow(z0, 2.0) * Math.pow(Math.PI, 4.0)))))) / t_1)));
	} else if (z0 <= -21000000000.0) {
		tmp = Math.atan((-z1 * ((1.0 + (-2.0 * Math.log(Math.pow(19333.689074365146, (z0 * z0))))) / t_1)));
	} else if (z0 <= 1.5e+14) {
		tmp = Math.atan((-z1 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / t_1)));
	} else {
		tmp = Math.atan((-z1 * (((Math.pow(z0, 2.0) * ((-1.0 * (Math.pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (Math.pow(Math.PI, 2.0) * t_2)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_2))) + (0.5 * (1.0 / (z2 * Math.PI)))) / z0)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = math.pow(math.pi, 3.0) / z2
	t_1 = -z2 * math.sin(((-2.0 * z0) * math.pi))
	t_2 = (math.pi / z2) - (0.3333333333333333 * (math.pi / z2))
	tmp = 0
	if z0 <= -1.35e+247:
		tmp = math.atan((-z1 * ((1.0 + (math.pow(z0, 2.0) * ((-2.0 * math.pow(math.pi, 2.0)) + (0.6666666666666666 * (math.pow(z0, 2.0) * math.pow(math.pi, 4.0)))))) / t_1)))
	elif z0 <= -21000000000.0:
		tmp = math.atan((-z1 * ((1.0 + (-2.0 * math.log(math.pow(19333.689074365146, (z0 * z0))))) / t_1)))
	elif z0 <= 1.5e+14:
		tmp = math.atan((-z1 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / t_1)))
	else:
		tmp = math.atan((-z1 * (((math.pow(z0, 2.0) * ((-1.0 * (math.pow(z0, 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * (math.pow(math.pi, 2.0) * t_2)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_2))) + (0.5 * (1.0 / (z2 * math.pi)))) / z0)))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64((pi ^ 3.0) / z2)
	t_1 = Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi)))
	t_2 = Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))
	tmp = 0.0
	if (z0 <= -1.35e+247)
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(1.0 + Float64((z0 ^ 2.0) * Float64(Float64(-2.0 * (pi ^ 2.0)) + Float64(0.6666666666666666 * Float64((z0 ^ 2.0) * (pi ^ 4.0)))))) / t_1)));
	elseif (z0 <= -21000000000.0)
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(1.0 + Float64(-2.0 * log((19333.689074365146 ^ Float64(z0 * z0))))) / t_1)));
	elseif (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(-z1) * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / t_1)));
	else
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64((z0 ^ 2.0) * Float64(Float64(-1.0 * Float64((z0 ^ 2.0) * Float64(Float64(-0.3333333333333333 * t_0) - Float64(Float64(-0.6666666666666666 * Float64((pi ^ 2.0) * t_2)) + Float64(-0.06666666666666667 * t_0))))) + Float64(-1.0 * t_2))) + Float64(0.5 * Float64(1.0 / Float64(z2 * pi)))) / z0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (pi ^ 3.0) / z2;
	t_1 = -z2 * sin(((-2.0 * z0) * pi));
	t_2 = (pi / z2) - (0.3333333333333333 * (pi / z2));
	tmp = 0.0;
	if (z0 <= -1.35e+247)
		tmp = atan((-z1 * ((1.0 + ((z0 ^ 2.0) * ((-2.0 * (pi ^ 2.0)) + (0.6666666666666666 * ((z0 ^ 2.0) * (pi ^ 4.0)))))) / t_1)));
	elseif (z0 <= -21000000000.0)
		tmp = atan((-z1 * ((1.0 + (-2.0 * log((19333.689074365146 ^ (z0 * z0))))) / t_1)));
	elseif (z0 <= 1.5e+14)
		tmp = atan((-z1 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / t_1)));
	else
		tmp = atan((-z1 * ((((z0 ^ 2.0) * ((-1.0 * ((z0 ^ 2.0) * ((-0.3333333333333333 * t_0) - ((-0.6666666666666666 * ((pi ^ 2.0) * t_2)) + (-0.06666666666666667 * t_0))))) + (-1.0 * t_2))) + (0.5 * (1.0 / (z2 * pi)))) / z0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[Power[Pi, 3.0], $MachinePrecision] / z2), $MachinePrecision]}, Block[{t$95$1 = N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1.35e+247], N[ArcTan[N[((-z1) * N[(N[(1.0 + N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, -21000000000.0], N[ArcTan[N[((-z1) * N[(N[(1.0 + N[(-2.0 * N[Log[N[Power[19333.689074365146, N[(z0 * z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.5e+14], N[ArcTan[N[((-z1) * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-z1) * N[(N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-1.0 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * t$95$0), $MachinePrecision] - N[(N[(-0.6666666666666666 * N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-0.06666666666666667 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < -1.35e247

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. add-log-exp N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\pi}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. pow-exp N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lower-exp.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   9. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{{z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(\color{blue}{-2 \cdot {\mathsf{PI}\left(\right)}^{2}} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{\frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{\frac{2}{3}} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       7. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{4}}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       10. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{4}}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       12. lower-PI.f64 74.2%

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   11. Applied rewrites 74.2%

       \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if -1.35e247 < z0 < -2.1e10

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. add-log-exp N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\pi}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. pow-exp N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lower-exp.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   9. Evaluated real constant 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if -2.1e10 < z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\pi}^{3}}{z2} - \left(\frac{-2}{3} \cdot \left({\pi}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right)\right) + \frac{-1}{15} \cdot \frac{{\pi}^{3}}{z2}\right)\right)\right) + -1 \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right)\right) + \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{{z0}^{2} \cdot \left(-1 \cdot \left({z0}^{2} \cdot \left(-0.3333333333333333 \cdot \frac{{\pi}^{3}}{z2} - \left(-0.6666666666666666 \cdot \left({\pi}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + -0.06666666666666667 \cdot \frac{{\pi}^{3}}{z2}\right)\right)\right) + -1 \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(-\left|z2\right|\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\\ t_1 := -\left|z1\right|\\ t_2 := \tan^{-1} \left(\frac{\left|z1\right|}{\left|z2\right|} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right)\\ t_3 := \left(z0 + z0\right) \cdot \pi\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z2\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(-\cos t\_3\right) \cdot \left|z1\right|}{\sin t\_3 \cdot \left|z2\right|}\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-208}:\\ \;\;\;\;\tan^{-1} \left(t\_1 \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}{t\_0}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-15}:\\ \;\;\;\;\tan^{-1} \left(t\_1 \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_1 \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{t\_0}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (- (fabs z2)) (sin (* (* -2.0 z0) PI))))
       (t_1 (- (fabs z1)))
       (t_2
        (atan
         (* (/ (fabs z1) (fabs z2)) (tan (* PI (+ 0.5 (+ z0 z0)))))))
       (t_3 (* (+ z0 z0) PI)))
  (*
   (copysign 1.0 z1)
   (*
    (copysign 1.0 z2)
    (if (<= t_2 -5e-9)
      (atan (/ (* (- (cos t_3)) (fabs z1)) (* (sin t_3) (fabs z2))))
      (if (<= t_2 -4e-208)
        (atan
         (*
          t_1
          (/
           (+
            1.0
            (*
             (pow z0 2.0)
             (+
              (* -2.0 (pow PI 2.0))
              (* 0.6666666666666666 (* (pow z0 2.0) (pow PI 4.0))))))
           t_0)))
        (if (<= t_2 1e-15)
          (atan
           (*
            t_1
            (/
             (+ 1.0 (* -2.0 (log (pow 19333.689074365146 (* z0 z0)))))
             t_0)))
          (atan
           (*
            t_1
            (/ (sin (+ (* (* PI -2.0) z0) (* 0.5 PI))) t_0))))))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = -fabs(z2) * sin(((-2.0 * z0) * ((double) M_PI)));
	double t_1 = -fabs(z1);
	double t_2 = atan(((fabs(z1) / fabs(z2)) * tan((((double) M_PI) * (0.5 + (z0 + z0))))));
	double t_3 = (z0 + z0) * ((double) M_PI);
	double tmp;
	if (t_2 <= -5e-9) {
		tmp = atan(((-cos(t_3) * fabs(z1)) / (sin(t_3) * fabs(z2))));
	} else if (t_2 <= -4e-208) {
		tmp = atan((t_1 * ((1.0 + (pow(z0, 2.0) * ((-2.0 * pow(((double) M_PI), 2.0)) + (0.6666666666666666 * (pow(z0, 2.0) * pow(((double) M_PI), 4.0)))))) / t_0)));
	} else if (t_2 <= 1e-15) {
		tmp = atan((t_1 * ((1.0 + (-2.0 * log(pow(19333.689074365146, (z0 * z0))))) / t_0)));
	} else {
		tmp = atan((t_1 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / t_0)));
	}
	return copysign(1.0, z1) * (copysign(1.0, z2) * tmp);
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = -Math.abs(z2) * Math.sin(((-2.0 * z0) * Math.PI));
	double t_1 = -Math.abs(z1);
	double t_2 = Math.atan(((Math.abs(z1) / Math.abs(z2)) * Math.tan((Math.PI * (0.5 + (z0 + z0))))));
	double t_3 = (z0 + z0) * Math.PI;
	double tmp;
	if (t_2 <= -5e-9) {
		tmp = Math.atan(((-Math.cos(t_3) * Math.abs(z1)) / (Math.sin(t_3) * Math.abs(z2))));
	} else if (t_2 <= -4e-208) {
		tmp = Math.atan((t_1 * ((1.0 + (Math.pow(z0, 2.0) * ((-2.0 * Math.pow(Math.PI, 2.0)) + (0.6666666666666666 * (Math.pow(z0, 2.0) * Math.pow(Math.PI, 4.0)))))) / t_0)));
	} else if (t_2 <= 1e-15) {
		tmp = Math.atan((t_1 * ((1.0 + (-2.0 * Math.log(Math.pow(19333.689074365146, (z0 * z0))))) / t_0)));
	} else {
		tmp = Math.atan((t_1 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / t_0)));
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z2) * tmp);
}

Python

def code(z1, z2, z0):
	t_0 = -math.fabs(z2) * math.sin(((-2.0 * z0) * math.pi))
	t_1 = -math.fabs(z1)
	t_2 = math.atan(((math.fabs(z1) / math.fabs(z2)) * math.tan((math.pi * (0.5 + (z0 + z0))))))
	t_3 = (z0 + z0) * math.pi
	tmp = 0
	if t_2 <= -5e-9:
		tmp = math.atan(((-math.cos(t_3) * math.fabs(z1)) / (math.sin(t_3) * math.fabs(z2))))
	elif t_2 <= -4e-208:
		tmp = math.atan((t_1 * ((1.0 + (math.pow(z0, 2.0) * ((-2.0 * math.pow(math.pi, 2.0)) + (0.6666666666666666 * (math.pow(z0, 2.0) * math.pow(math.pi, 4.0)))))) / t_0)))
	elif t_2 <= 1e-15:
		tmp = math.atan((t_1 * ((1.0 + (-2.0 * math.log(math.pow(19333.689074365146, (z0 * z0))))) / t_0)))
	else:
		tmp = math.atan((t_1 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / t_0)))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z2) * tmp)

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(-abs(z2)) * sin(Float64(Float64(-2.0 * z0) * pi)))
	t_1 = Float64(-abs(z1))
	t_2 = atan(Float64(Float64(abs(z1) / abs(z2)) * tan(Float64(pi * Float64(0.5 + Float64(z0 + z0))))))
	t_3 = Float64(Float64(z0 + z0) * pi)
	tmp = 0.0
	if (t_2 <= -5e-9)
		tmp = atan(Float64(Float64(Float64(-cos(t_3)) * abs(z1)) / Float64(sin(t_3) * abs(z2))));
	elseif (t_2 <= -4e-208)
		tmp = atan(Float64(t_1 * Float64(Float64(1.0 + Float64((z0 ^ 2.0) * Float64(Float64(-2.0 * (pi ^ 2.0)) + Float64(0.6666666666666666 * Float64((z0 ^ 2.0) * (pi ^ 4.0)))))) / t_0)));
	elseif (t_2 <= 1e-15)
		tmp = atan(Float64(t_1 * Float64(Float64(1.0 + Float64(-2.0 * log((19333.689074365146 ^ Float64(z0 * z0))))) / t_0)));
	else
		tmp = atan(Float64(t_1 * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / t_0)));
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z2) * tmp))
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = -abs(z2) * sin(((-2.0 * z0) * pi));
	t_1 = -abs(z1);
	t_2 = atan(((abs(z1) / abs(z2)) * tan((pi * (0.5 + (z0 + z0))))));
	t_3 = (z0 + z0) * pi;
	tmp = 0.0;
	if (t_2 <= -5e-9)
		tmp = atan(((-cos(t_3) * abs(z1)) / (sin(t_3) * abs(z2))));
	elseif (t_2 <= -4e-208)
		tmp = atan((t_1 * ((1.0 + ((z0 ^ 2.0) * ((-2.0 * (pi ^ 2.0)) + (0.6666666666666666 * ((z0 ^ 2.0) * (pi ^ 4.0)))))) / t_0)));
	elseif (t_2 <= 1e-15)
		tmp = atan((t_1 * ((1.0 + (-2.0 * log((19333.689074365146 ^ (z0 * z0))))) / t_0)));
	else
		tmp = atan((t_1 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / t_0)));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z2) * abs(1.0)) * tmp);
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[((-N[Abs[z2], $MachinePrecision]) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[z1], $MachinePrecision])}, Block[{t$95$2 = N[ArcTan[N[(N[(N[Abs[z1], $MachinePrecision] / N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * N[(0.5 + N[(z0 + z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(z0 + z0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -5e-9], N[ArcTan[N[(N[((-N[Cos[t$95$3], $MachinePrecision]) * N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$3], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, -4e-208], N[ArcTan[N[(t$95$1 * N[(N[(1.0 + N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e-15], N[ArcTan[N[(t$95$1 * N[(N[(1.0 + N[(-2.0 * N[Log[N[Power[19333.689074365146, N[(z0 * z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$1 * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (atan.f64 (*.f64 (/.f64 z1 z2) (tan.f64 (*.f64 (PI.f64) (+.f64 #s(literal 1/2 binary64) (+.f64 z0 z0)))))) < -5.0000000000000001e-9

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \frac{z1}{z2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)} \cdot \frac{z1}{z2}\right) \]

      4. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}} \cdot \frac{z1}{z2}\right) \]

      5. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}{\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}} \cdot \frac{z1}{z2}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}{\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \cdot \color{blue}{\frac{z1}{z2}}\right) \]

      7. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z1}{\left(\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z2}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z1}{\left(\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z2}\right)} \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-\cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot z1}{\sin \left(\left(z0 + z0\right) \cdot \pi\right) \cdot z2}\right)} \]

   if -5.0000000000000001e-9 < (atan.f64 (*.f64 (/.f64 z1 z2) (tan.f64 (*.f64 (PI.f64) (+.f64 #s(literal 1/2 binary64) (+.f64 z0 z0)))))) < -4.0000000000000004e-208

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. add-log-exp N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\pi}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. pow-exp N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lower-exp.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   9. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{{z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(\color{blue}{-2 \cdot {\mathsf{PI}\left(\right)}^{2}} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{\frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{\frac{2}{3}} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       7. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{4}}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       10. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + \frac{2}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{4}}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

       12. lower-PI.f64 74.2%

           \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   11. Applied rewrites 74.2%

       \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if -4.0000000000000004e-208 < (atan.f64 (*.f64 (/.f64 z1 z2) (tan.f64 (*.f64 (PI.f64) (+.f64 #s(literal 1/2 binary64) (+.f64 z0 z0)))))) < 1.0000000000000001e-15

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. add-log-exp N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\pi}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. pow-exp N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lower-exp.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   9. Evaluated real constant 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.0000000000000001e-15 < (atan.f64 (*.f64 (/.f64 z1 z2) (tan.f64 (*.f64 (PI.f64) (+.f64 #s(literal 1/2 binary64) (+.f64 z0 z0))))))

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\left|z1\right|\\ t_1 := \left(-\left|z2\right|\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z2\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|z1\right|}{\left|z2\right|} \leq 140000000:\\ \;\;\;\;\tan^{-1} \left(t\_0 \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(t\_0 \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{t\_1}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (- (fabs z1)))
       (t_1 (* (- (fabs z2)) (sin (* (* -2.0 z0) PI)))))
  (*
   (copysign 1.0 z1)
   (*
    (copysign 1.0 z2)
    (if (<= (/ (fabs z1) (fabs z2)) 140000000.0)
      (atan
       (*
        t_0
        (/
         (+ 1.0 (* -2.0 (log (pow 19333.689074365146 (* z0 z0)))))
         t_1)))
      (atan
       (* t_0 (/ (sin (+ (* (* PI -2.0) z0) (* 0.5 PI))) t_1))))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = -fabs(z1);
	double t_1 = -fabs(z2) * sin(((-2.0 * z0) * ((double) M_PI)));
	double tmp;
	if ((fabs(z1) / fabs(z2)) <= 140000000.0) {
		tmp = atan((t_0 * ((1.0 + (-2.0 * log(pow(19333.689074365146, (z0 * z0))))) / t_1)));
	} else {
		tmp = atan((t_0 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / t_1)));
	}
	return copysign(1.0, z1) * (copysign(1.0, z2) * tmp);
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = -Math.abs(z1);
	double t_1 = -Math.abs(z2) * Math.sin(((-2.0 * z0) * Math.PI));
	double tmp;
	if ((Math.abs(z1) / Math.abs(z2)) <= 140000000.0) {
		tmp = Math.atan((t_0 * ((1.0 + (-2.0 * Math.log(Math.pow(19333.689074365146, (z0 * z0))))) / t_1)));
	} else {
		tmp = Math.atan((t_0 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / t_1)));
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z2) * tmp);
}

Python

def code(z1, z2, z0):
	t_0 = -math.fabs(z1)
	t_1 = -math.fabs(z2) * math.sin(((-2.0 * z0) * math.pi))
	tmp = 0
	if (math.fabs(z1) / math.fabs(z2)) <= 140000000.0:
		tmp = math.atan((t_0 * ((1.0 + (-2.0 * math.log(math.pow(19333.689074365146, (z0 * z0))))) / t_1)))
	else:
		tmp = math.atan((t_0 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / t_1)))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z2) * tmp)

Julia

function code(z1, z2, z0)
	t_0 = Float64(-abs(z1))
	t_1 = Float64(Float64(-abs(z2)) * sin(Float64(Float64(-2.0 * z0) * pi)))
	tmp = 0.0
	if (Float64(abs(z1) / abs(z2)) <= 140000000.0)
		tmp = atan(Float64(t_0 * Float64(Float64(1.0 + Float64(-2.0 * log((19333.689074365146 ^ Float64(z0 * z0))))) / t_1)));
	else
		tmp = atan(Float64(t_0 * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / t_1)));
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z2) * tmp))
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = -abs(z1);
	t_1 = -abs(z2) * sin(((-2.0 * z0) * pi));
	tmp = 0.0;
	if ((abs(z1) / abs(z2)) <= 140000000.0)
		tmp = atan((t_0 * ((1.0 + (-2.0 * log((19333.689074365146 ^ (z0 * z0))))) / t_1)));
	else
		tmp = atan((t_0 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / t_1)));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z2) * abs(1.0)) * tmp);
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = (-N[Abs[z1], $MachinePrecision])}, Block[{t$95$1 = N[((-N[Abs[z2], $MachinePrecision]) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Abs[z1], $MachinePrecision] / N[Abs[z2], $MachinePrecision]), $MachinePrecision], 140000000.0], N[ArcTan[N[(t$95$0 * N[(N[(1.0 + N[(-2.0 * N[Log[N[Power[19333.689074365146, N[(z0 * z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$0 * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z1 z2) < 1.4e8

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. add-log-exp N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \left(\pi \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. log-pow-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-log.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left({\left(e^{\pi}\right)}^{\pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. pow-exp N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lower-exp.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{{\pi}^{2}}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left({z0}^{2}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 74.6%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({\left(e^{\pi \cdot \pi}\right)}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   9. Evaluated real constant 74.6%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \log \left({19333.689074365146}^{\left(z0 \cdot z0\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.4e8 < (/.f64 z1 z2)

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\\ t_1 := \left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;z0 \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\tan^{-1} \left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)\\ \mathbf{elif}\;z0 \leq -8500000000:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{t\_1 \cdot t\_1 - 1 \cdot 1}{t\_1 - 1}}{t\_0}\right)\\ \mathbf{elif}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (- z2) (sin (* (* -2.0 z0) PI))))
       (t_1 (* (* (* (* PI PI) -2.0) z0) z0)))
  (if (<= z0 -2.7e+106)
    (atan (* z1 (/ (tan (* 0.5 PI)) z2)))
    (if (<= z0 -8500000000.0)
      (atan
       (* (- z1) (/ (/ (- (* t_1 t_1) (* 1.0 1.0)) (- t_1 1.0)) t_0)))
      (if (<= z0 1.5e+14)
        (atan (* (- z1) (/ (cos (* (+ z0 z0) PI)) t_0)))
        (atan
         (/
          (*
           z1
           (-
            (*
             (pow z0 2.0)
             (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
            (* 0.5 (/ 1.0 (* z2 PI)))))
          z0)))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = -z2 * sin(((-2.0 * z0) * ((double) M_PI)));
	double t_1 = (((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0;
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = atan((z1 * (tan((0.5 * ((double) M_PI))) / z2)));
	} else if (z0 <= -8500000000.0) {
		tmp = atan((-z1 * ((((t_1 * t_1) - (1.0 * 1.0)) / (t_1 - 1.0)) / t_0)));
	} else if (z0 <= 1.5e+14) {
		tmp = atan((-z1 * (cos(((z0 + z0) * ((double) M_PI))) / t_0)));
	} else {
		tmp = atan(((z1 * ((pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2)))) - (0.5 * (1.0 / (z2 * ((double) M_PI)))))) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = -z2 * Math.sin(((-2.0 * z0) * Math.PI));
	double t_1 = (((Math.PI * Math.PI) * -2.0) * z0) * z0;
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = Math.atan((z1 * (Math.tan((0.5 * Math.PI)) / z2)));
	} else if (z0 <= -8500000000.0) {
		tmp = Math.atan((-z1 * ((((t_1 * t_1) - (1.0 * 1.0)) / (t_1 - 1.0)) / t_0)));
	} else if (z0 <= 1.5e+14) {
		tmp = Math.atan((-z1 * (Math.cos(((z0 + z0) * Math.PI)) / t_0)));
	} else {
		tmp = Math.atan(((z1 * ((Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2)))) - (0.5 * (1.0 / (z2 * Math.PI))))) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = -z2 * math.sin(((-2.0 * z0) * math.pi))
	t_1 = (((math.pi * math.pi) * -2.0) * z0) * z0
	tmp = 0
	if z0 <= -2.7e+106:
		tmp = math.atan((z1 * (math.tan((0.5 * math.pi)) / z2)))
	elif z0 <= -8500000000.0:
		tmp = math.atan((-z1 * ((((t_1 * t_1) - (1.0 * 1.0)) / (t_1 - 1.0)) / t_0)))
	elif z0 <= 1.5e+14:
		tmp = math.atan((-z1 * (math.cos(((z0 + z0) * math.pi)) / t_0)))
	else:
		tmp = math.atan(((z1 * ((math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2)))) - (0.5 * (1.0 / (z2 * math.pi))))) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi)))
	t_1 = Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0)
	tmp = 0.0
	if (z0 <= -2.7e+106)
		tmp = atan(Float64(z1 * Float64(tan(Float64(0.5 * pi)) / z2)));
	elseif (z0 <= -8500000000.0)
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64(Float64(t_1 * t_1) - Float64(1.0 * 1.0)) / Float64(t_1 - 1.0)) / t_0)));
	elseif (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(-z1) * Float64(cos(Float64(Float64(z0 + z0) * pi)) / t_0)));
	else
		tmp = atan(Float64(Float64(z1 * Float64(Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))) - Float64(0.5 * Float64(1.0 / Float64(z2 * pi))))) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = -z2 * sin(((-2.0 * z0) * pi));
	t_1 = (((pi * pi) * -2.0) * z0) * z0;
	tmp = 0.0;
	if (z0 <= -2.7e+106)
		tmp = atan((z1 * (tan((0.5 * pi)) / z2)));
	elseif (z0 <= -8500000000.0)
		tmp = atan((-z1 * ((((t_1 * t_1) - (1.0 * 1.0)) / (t_1 - 1.0)) / t_0)));
	elseif (z0 <= 1.5e+14)
		tmp = atan((-z1 * (cos(((z0 + z0) * pi)) / t_0)));
	else
		tmp = atan(((z1 * (((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2)))) - (0.5 * (1.0 / (z2 * pi))))) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[z0, -2.7e+106], N[ArcTan[N[(z1 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, -8500000000.0], N[ArcTan[N[((-z1) * N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.5e+14], N[ArcTan[N[((-z1) * N[(N[Cos[N[(N[(z0 + z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(z1 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < -2.7000000000000001e106

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{0.5}\right)\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      6. lower-/.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot 0.5\right)}{z2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}{z2}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}}{z2}\right) \]

      9. lower-*.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(0.5 \cdot \pi\right)}}{z2}\right) \]

   6. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)} \]

   if -2.7000000000000001e106 < z0 < -8.5e9

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) + \color{blue}{1}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. flip-+ N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 60.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) - 1 \cdot 1}{\color{blue}{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if -8.5e9 < z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

   9. Applied rewrites 67.5%

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\\ t_1 := \left(z0 + z0\right) \cdot \pi\\ \mathbf{if}\;z0 \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\tan^{-1} \left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)\\ \mathbf{elif}\;z0 \leq -8500000000:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{t\_0 \cdot t\_0 - 1 \cdot 1}{t\_0 - 1}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\\ \mathbf{elif}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(-\cos t\_1\right) \cdot z1}{\sin t\_1 \cdot z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (* (* (* PI PI) -2.0) z0) z0)) (t_1 (* (+ z0 z0) PI)))
  (if (<= z0 -2.7e+106)
    (atan (* z1 (/ (tan (* 0.5 PI)) z2)))
    (if (<= z0 -8500000000.0)
      (atan
       (*
        (- z1)
        (/
         (/ (- (* t_0 t_0) (* 1.0 1.0)) (- t_0 1.0))
         (* (- z2) (sin (* (* -2.0 z0) PI))))))
      (if (<= z0 1.5e+14)
        (atan (/ (* (- (cos t_1)) z1) (* (sin t_1) z2)))
        (atan
         (/
          (*
           z1
           (-
            (*
             (pow z0 2.0)
             (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
            (* 0.5 (/ 1.0 (* z2 PI)))))
          z0)))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0;
	double t_1 = (z0 + z0) * ((double) M_PI);
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = atan((z1 * (tan((0.5 * ((double) M_PI))) / z2)));
	} else if (z0 <= -8500000000.0) {
		tmp = atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * sin(((-2.0 * z0) * ((double) M_PI)))))));
	} else if (z0 <= 1.5e+14) {
		tmp = atan(((-cos(t_1) * z1) / (sin(t_1) * z2)));
	} else {
		tmp = atan(((z1 * ((pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2)))) - (0.5 * (1.0 / (z2 * ((double) M_PI)))))) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (((Math.PI * Math.PI) * -2.0) * z0) * z0;
	double t_1 = (z0 + z0) * Math.PI;
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = Math.atan((z1 * (Math.tan((0.5 * Math.PI)) / z2)));
	} else if (z0 <= -8500000000.0) {
		tmp = Math.atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * Math.sin(((-2.0 * z0) * Math.PI))))));
	} else if (z0 <= 1.5e+14) {
		tmp = Math.atan(((-Math.cos(t_1) * z1) / (Math.sin(t_1) * z2)));
	} else {
		tmp = Math.atan(((z1 * ((Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2)))) - (0.5 * (1.0 / (z2 * Math.PI))))) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (((math.pi * math.pi) * -2.0) * z0) * z0
	t_1 = (z0 + z0) * math.pi
	tmp = 0
	if z0 <= -2.7e+106:
		tmp = math.atan((z1 * (math.tan((0.5 * math.pi)) / z2)))
	elif z0 <= -8500000000.0:
		tmp = math.atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * math.sin(((-2.0 * z0) * math.pi))))))
	elif z0 <= 1.5e+14:
		tmp = math.atan(((-math.cos(t_1) * z1) / (math.sin(t_1) * z2)))
	else:
		tmp = math.atan(((z1 * ((math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2)))) - (0.5 * (1.0 / (z2 * math.pi))))) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0)
	t_1 = Float64(Float64(z0 + z0) * pi)
	tmp = 0.0
	if (z0 <= -2.7e+106)
		tmp = atan(Float64(z1 * Float64(tan(Float64(0.5 * pi)) / z2)));
	elseif (z0 <= -8500000000.0)
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(1.0 * 1.0)) / Float64(t_0 - 1.0)) / Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi))))));
	elseif (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(Float64(-cos(t_1)) * z1) / Float64(sin(t_1) * z2)));
	else
		tmp = atan(Float64(Float64(z1 * Float64(Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))) - Float64(0.5 * Float64(1.0 / Float64(z2 * pi))))) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (((pi * pi) * -2.0) * z0) * z0;
	t_1 = (z0 + z0) * pi;
	tmp = 0.0;
	if (z0 <= -2.7e+106)
		tmp = atan((z1 * (tan((0.5 * pi)) / z2)));
	elseif (z0 <= -8500000000.0)
		tmp = atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * sin(((-2.0 * z0) * pi))))));
	elseif (z0 <= 1.5e+14)
		tmp = atan(((-cos(t_1) * z1) / (sin(t_1) * z2)));
	else
		tmp = atan(((z1 * (((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2)))) - (0.5 * (1.0 / (z2 * pi))))) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z0 + z0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[z0, -2.7e+106], N[ArcTan[N[(z1 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, -8500000000.0], N[ArcTan[N[((-z1) * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] / N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.5e+14], N[ArcTan[N[(N[((-N[Cos[t$95$1], $MachinePrecision]) * z1), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(z1 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < -2.7000000000000001e106

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{0.5}\right)\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      6. lower-/.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot 0.5\right)}{z2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}{z2}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}}{z2}\right) \]

      9. lower-*.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(0.5 \cdot \pi\right)}}{z2}\right) \]

   6. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)} \]

   if -2.7000000000000001e106 < z0 < -8.5e9

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) + \color{blue}{1}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. flip-+ N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 60.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) - 1 \cdot 1}{\color{blue}{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if -8.5e9 < z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \frac{z1}{z2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)} \cdot \frac{z1}{z2}\right) \]

      4. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}} \cdot \frac{z1}{z2}\right) \]

      5. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}{\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}} \cdot \frac{z1}{z2}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)}{\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \cdot \color{blue}{\frac{z1}{z2}}\right) \]

      7. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z1}{\left(\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z2}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z1}{\left(\mathsf{neg}\left(\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)\right) \cdot z2}\right)} \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-\cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot z1}{\sin \left(\left(z0 + z0\right) \cdot \pi\right) \cdot z2}\right)} \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

   9. Applied rewrites 67.5%

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\\ t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z2\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|z1\right|}{\left|z2\right|} \leq 140000000:\\ \;\;\;\;\tan^{-1} \left(\frac{\left|z1\right| \cdot \left(\left(\left(\left(-0.08888888888888889 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(t\_1 \cdot t\_1\right) - -0.6666666666666666 \cdot {\pi}^{4}\right) \cdot \left(z0 \cdot z0\right) - 2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(z0 \cdot z0\right) - -1\right)}{t\_0 \cdot \left|z2\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\left(-\left|z1\right|\right) \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{\left(-\left|z2\right|\right) \cdot t\_0}\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (sin (* (* -2.0 z0) PI))) (t_1 (* (* PI PI) PI)))
  (*
   (copysign 1.0 z1)
   (*
    (copysign 1.0 z2)
    (if (<= (/ (fabs z1) (fabs z2)) 140000000.0)
      (atan
       (/
        (*
         (fabs z1)
         (-
          (*
           (-
            (*
             (-
              (* (* -0.08888888888888889 (* z0 z0)) (* t_1 t_1))
              (* -0.6666666666666666 (pow PI 4.0)))
             (* z0 z0))
            (* 2.0 (* PI PI)))
           (* z0 z0))
          -1.0))
        (* t_0 (fabs z2))))
      (atan
       (*
        (- (fabs z1))
        (/
         (sin (+ (* (* PI -2.0) z0) (* 0.5 PI)))
         (* (- (fabs z2)) t_0)))))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = sin(((-2.0 * z0) * ((double) M_PI)));
	double t_1 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double tmp;
	if ((fabs(z1) / fabs(z2)) <= 140000000.0) {
		tmp = atan(((fabs(z1) * (((((((-0.08888888888888889 * (z0 * z0)) * (t_1 * t_1)) - (-0.6666666666666666 * pow(((double) M_PI), 4.0))) * (z0 * z0)) - (2.0 * (((double) M_PI) * ((double) M_PI)))) * (z0 * z0)) - -1.0)) / (t_0 * fabs(z2))));
	} else {
		tmp = atan((-fabs(z1) * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / (-fabs(z2) * t_0))));
	}
	return copysign(1.0, z1) * (copysign(1.0, z2) * tmp);
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = Math.sin(((-2.0 * z0) * Math.PI));
	double t_1 = (Math.PI * Math.PI) * Math.PI;
	double tmp;
	if ((Math.abs(z1) / Math.abs(z2)) <= 140000000.0) {
		tmp = Math.atan(((Math.abs(z1) * (((((((-0.08888888888888889 * (z0 * z0)) * (t_1 * t_1)) - (-0.6666666666666666 * Math.pow(Math.PI, 4.0))) * (z0 * z0)) - (2.0 * (Math.PI * Math.PI))) * (z0 * z0)) - -1.0)) / (t_0 * Math.abs(z2))));
	} else {
		tmp = Math.atan((-Math.abs(z1) * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / (-Math.abs(z2) * t_0))));
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z2) * tmp);
}

Python

def code(z1, z2, z0):
	t_0 = math.sin(((-2.0 * z0) * math.pi))
	t_1 = (math.pi * math.pi) * math.pi
	tmp = 0
	if (math.fabs(z1) / math.fabs(z2)) <= 140000000.0:
		tmp = math.atan(((math.fabs(z1) * (((((((-0.08888888888888889 * (z0 * z0)) * (t_1 * t_1)) - (-0.6666666666666666 * math.pow(math.pi, 4.0))) * (z0 * z0)) - (2.0 * (math.pi * math.pi))) * (z0 * z0)) - -1.0)) / (t_0 * math.fabs(z2))))
	else:
		tmp = math.atan((-math.fabs(z1) * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / (-math.fabs(z2) * t_0))))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z2) * tmp)

Julia

function code(z1, z2, z0)
	t_0 = sin(Float64(Float64(-2.0 * z0) * pi))
	t_1 = Float64(Float64(pi * pi) * pi)
	tmp = 0.0
	if (Float64(abs(z1) / abs(z2)) <= 140000000.0)
		tmp = atan(Float64(Float64(abs(z1) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.08888888888888889 * Float64(z0 * z0)) * Float64(t_1 * t_1)) - Float64(-0.6666666666666666 * (pi ^ 4.0))) * Float64(z0 * z0)) - Float64(2.0 * Float64(pi * pi))) * Float64(z0 * z0)) - -1.0)) / Float64(t_0 * abs(z2))));
	else
		tmp = atan(Float64(Float64(-abs(z1)) * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / Float64(Float64(-abs(z2)) * t_0))));
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z2) * tmp))
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = sin(((-2.0 * z0) * pi));
	t_1 = (pi * pi) * pi;
	tmp = 0.0;
	if ((abs(z1) / abs(z2)) <= 140000000.0)
		tmp = atan(((abs(z1) * (((((((-0.08888888888888889 * (z0 * z0)) * (t_1 * t_1)) - (-0.6666666666666666 * (pi ^ 4.0))) * (z0 * z0)) - (2.0 * (pi * pi))) * (z0 * z0)) - -1.0)) / (t_0 * abs(z2))));
	else
		tmp = atan((-abs(z1) * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / (-abs(z2) * t_0))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z2) * abs(1.0)) * tmp);
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Abs[z1], $MachinePrecision] / N[Abs[z2], $MachinePrecision]), $MachinePrecision], 140000000.0], N[ArcTan[N[(N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-0.08888888888888889 * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-0.6666666666666666 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-N[Abs[z1], $MachinePrecision]) * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-N[Abs[z2], $MachinePrecision]) * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z1 z2) < 1.4e8

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + {z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\pi}^{6}\right) + \frac{2}{3} \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{{z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(\color{blue}{-2 \cdot {\mathsf{PI}\left(\right)}^{2}} + {z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{{z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{{z0}^{2}} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {z0}^{\color{blue}{2}} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + {z0}^{2} \cdot \left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + {z0}^{2} \cdot \color{blue}{\left(\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + {z0}^{2} \cdot \left(\color{blue}{\frac{-4}{45} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)} + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 74.1%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + {z0}^{2} \cdot \left(-0.08888888888888889 \cdot \left({z0}^{2} \cdot {\pi}^{6}\right) + 0.6666666666666666 \cdot {\pi}^{4}\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 74.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \left(\left(\left(\left(-0.08888888888888889 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) - -0.6666666666666666 \cdot {\pi}^{4}\right) \cdot \left(z0 \cdot z0\right) - 2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(z0 \cdot z0\right) - -1\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot z2}\right)} \]

   if 1.4e8 < (/.f64 z1 z2)

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;z0 \leq -2.7 \cdot 10^{+106}:\\ \;\;\;\;\tan^{-1} \left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)\\ \mathbf{elif}\;z0 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{t\_0 \cdot t\_0 - 1 \cdot 1}{t\_0 - 1}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (* (* (* PI PI) -2.0) z0) z0)))
  (if (<= z0 -2.7e+106)
    (atan (* z1 (/ (tan (* 0.5 PI)) z2)))
    (if (<= z0 2e-46)
      (atan
       (*
        (- z1)
        (/
         (/ (- (* t_0 t_0) (* 1.0 1.0)) (- t_0 1.0))
         (* (- z2) (sin (* (* -2.0 z0) PI))))))
      (atan
       (/
        (*
         z1
         (-
          (*
           (pow z0 2.0)
           (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
          (* 0.5 (/ 1.0 (* z2 PI)))))
        z0))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0;
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = atan((z1 * (tan((0.5 * ((double) M_PI))) / z2)));
	} else if (z0 <= 2e-46) {
		tmp = atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * sin(((-2.0 * z0) * ((double) M_PI)))))));
	} else {
		tmp = atan(((z1 * ((pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2)))) - (0.5 * (1.0 / (z2 * ((double) M_PI)))))) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (((Math.PI * Math.PI) * -2.0) * z0) * z0;
	double tmp;
	if (z0 <= -2.7e+106) {
		tmp = Math.atan((z1 * (Math.tan((0.5 * Math.PI)) / z2)));
	} else if (z0 <= 2e-46) {
		tmp = Math.atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * Math.sin(((-2.0 * z0) * Math.PI))))));
	} else {
		tmp = Math.atan(((z1 * ((Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2)))) - (0.5 * (1.0 / (z2 * Math.PI))))) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (((math.pi * math.pi) * -2.0) * z0) * z0
	tmp = 0
	if z0 <= -2.7e+106:
		tmp = math.atan((z1 * (math.tan((0.5 * math.pi)) / z2)))
	elif z0 <= 2e-46:
		tmp = math.atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * math.sin(((-2.0 * z0) * math.pi))))))
	else:
		tmp = math.atan(((z1 * ((math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2)))) - (0.5 * (1.0 / (z2 * math.pi))))) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0)
	tmp = 0.0
	if (z0 <= -2.7e+106)
		tmp = atan(Float64(z1 * Float64(tan(Float64(0.5 * pi)) / z2)));
	elseif (z0 <= 2e-46)
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(1.0 * 1.0)) / Float64(t_0 - 1.0)) / Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi))))));
	else
		tmp = atan(Float64(Float64(z1 * Float64(Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))) - Float64(0.5 * Float64(1.0 / Float64(z2 * pi))))) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (((pi * pi) * -2.0) * z0) * z0;
	tmp = 0.0;
	if (z0 <= -2.7e+106)
		tmp = atan((z1 * (tan((0.5 * pi)) / z2)));
	elseif (z0 <= 2e-46)
		tmp = atan((-z1 * ((((t_0 * t_0) - (1.0 * 1.0)) / (t_0 - 1.0)) / (-z2 * sin(((-2.0 * z0) * pi))))));
	else
		tmp = atan(((z1 * (((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2)))) - (0.5 * (1.0 / (z2 * pi))))) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[z0, -2.7e+106], N[ArcTan[N[(z1 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 2e-46], N[ArcTan[N[((-z1) * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] / N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(z1 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -2.7000000000000001e106

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{0.5}\right)\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      6. lower-/.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot 0.5\right)}{z2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}{z2}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}}{z2}\right) \]

      9. lower-*.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(0.5 \cdot \pi\right)}}{z2}\right) \]

   6. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)} \]

   if -2.7000000000000001e106 < z0 < 2e-46

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) + \color{blue}{1}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. flip-+ N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right) - 1 \cdot 1}{\color{blue}{-2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right) - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 60.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\frac{\left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0\right) - 1 \cdot 1}{\color{blue}{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - 1}}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 2e-46 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

   9. Applied rewrites 67.5%

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 1.5e+14)
  (atan
   (*
    (- z1)
    (/
     (sin (+ (* (* PI -2.0) z0) (* 0.5 PI)))
     (* (- z2) (sin (* (* -2.0 z0) PI))))))
  (atan
   (/
    (*
     z1
     (-
      (* (pow z0 2.0) (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
      (* 0.5 (/ 1.0 (* z2 PI)))))
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.5e+14) {
		tmp = atan((-z1 * (sin((((((double) M_PI) * -2.0) * z0) + (0.5 * ((double) M_PI)))) / (-z2 * sin(((-2.0 * z0) * ((double) M_PI)))))));
	} else {
		tmp = atan(((z1 * ((pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2)))) - (0.5 * (1.0 / (z2 * ((double) M_PI)))))) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.5e+14) {
		tmp = Math.atan((-z1 * (Math.sin((((Math.PI * -2.0) * z0) + (0.5 * Math.PI))) / (-z2 * Math.sin(((-2.0 * z0) * Math.PI))))));
	} else {
		tmp = Math.atan(((z1 * ((Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2)))) - (0.5 * (1.0 / (z2 * Math.PI))))) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 1.5e+14:
		tmp = math.atan((-z1 * (math.sin((((math.pi * -2.0) * z0) + (0.5 * math.pi))) / (-z2 * math.sin(((-2.0 * z0) * math.pi))))))
	else:
		tmp = math.atan(((z1 * ((math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2)))) - (0.5 * (1.0 / (z2 * math.pi))))) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 1.5e+14)
		tmp = atan(Float64(Float64(-z1) * Float64(sin(Float64(Float64(Float64(pi * -2.0) * z0) + Float64(0.5 * pi))) / Float64(Float64(-z2) * sin(Float64(Float64(-2.0 * z0) * pi))))));
	else
		tmp = atan(Float64(Float64(z1 * Float64(Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))) - Float64(0.5 * Float64(1.0 / Float64(z2 * pi))))) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 1.5e+14)
		tmp = atan((-z1 * (sin((((pi * -2.0) * z0) + (0.5 * pi))) / (-z2 * sin(((-2.0 * z0) * pi))))));
	else
		tmp = atan(((z1 * (((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2)))) - (0.5 * (1.0 / (z2 * pi))))) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 1.5e+14], N[ArcTan[N[((-z1) * N[(N[Sin[N[(N[(N[(Pi * -2.0), $MachinePrecision] * z0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-z2) * N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(z1 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.5e14

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{\color{blue}{\pi}}{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. mult-flip N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(z0 + z0\right) \cdot \pi\right)\right) + \frac{1}{2} \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right) \cdot \pi}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      12. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\pi \cdot \left(\mathsf{neg}\left(\left(z0 + z0\right)\right)\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{\left(z0 + z0\right)}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      15. count-2 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot z0}\right)\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot z0\right)} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      17. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\color{blue}{-2} \cdot z0\right) + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      18. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right) \cdot z0} + \frac{1}{2} \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      20. lower-*.f64 76.7%

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\color{blue}{\left(\pi \cdot -2\right)} \cdot z0 + 0.5 \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   5. Applied rewrites 76.7%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{\sin \left(\left(\pi \cdot -2\right) \cdot z0 + 0.5 \cdot \pi\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   if 1.5e14 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

   9. Applied rewrites 67.5%

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1} \left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\left(-z1\right) \cdot \frac{-1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 -5.6e+15)
  (atan (* z1 (/ (tan (* 0.5 PI)) z2)))
  (atan
   (*
    (- z1)
    (/
     (+
      (*
       -1.0
       (*
        (pow z0 2.0)
        (- (/ PI z2) (* 0.3333333333333333 (/ PI z2)))))
      (* 0.5 (/ 1.0 (* z2 PI))))
     z0)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= -5.6e+15) {
		tmp = atan((z1 * (tan((0.5 * ((double) M_PI))) / z2)));
	} else {
		tmp = atan((-z1 * (((-1.0 * (pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2))))) + (0.5 * (1.0 / (z2 * ((double) M_PI))))) / z0)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= -5.6e+15) {
		tmp = Math.atan((z1 * (Math.tan((0.5 * Math.PI)) / z2)));
	} else {
		tmp = Math.atan((-z1 * (((-1.0 * (Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2))))) + (0.5 * (1.0 / (z2 * Math.PI)))) / z0)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= -5.6e+15:
		tmp = math.atan((z1 * (math.tan((0.5 * math.pi)) / z2)))
	else:
		tmp = math.atan((-z1 * (((-1.0 * (math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2))))) + (0.5 * (1.0 / (z2 * math.pi)))) / z0)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= -5.6e+15)
		tmp = atan(Float64(z1 * Float64(tan(Float64(0.5 * pi)) / z2)));
	else
		tmp = atan(Float64(Float64(-z1) * Float64(Float64(Float64(-1.0 * Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2))))) + Float64(0.5 * Float64(1.0 / Float64(z2 * pi)))) / z0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= -5.6e+15)
		tmp = atan((z1 * (tan((0.5 * pi)) / z2)));
	else
		tmp = atan((-z1 * (((-1.0 * ((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2))))) + (0.5 * (1.0 / (z2 * pi)))) / z0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, -5.6e+15], N[ArcTan[N[(z1 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[((-z1) * N[(N[(N[(-1.0 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -5.6e15

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{0.5}\right)\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      6. lower-/.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot 0.5\right)}{z2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}{z2}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}}{z2}\right) \]

      9. lower-*.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(0.5 \cdot \pi\right)}}{z2}\right) \]

   6. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)} \]

   if -5.6e15 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{-1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right)\right) + \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{-1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right)\right) + \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{-1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right)\right) + 0.5 \cdot \frac{1}{z2 \cdot \pi}}{z0}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 2e-46)
  (-
   (atan
    (*
     (/
      (- (* (* (* (* PI PI) -2.0) z0) z0) -1.0)
      (* (sin (* (* -2.0 z0) PI)) (- z2)))
     z1)))
  (atan
   (/
    (*
     z1
     (-
      (* (pow z0 2.0) (- (/ PI z2) (* 0.3333333333333333 (/ PI z2))))
      (* 0.5 (/ 1.0 (* z2 PI)))))
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 2e-46) {
		tmp = -atan((((((((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * ((double) M_PI))) * -z2)) * z1));
	} else {
		tmp = atan(((z1 * ((pow(z0, 2.0) * ((((double) M_PI) / z2) - (0.3333333333333333 * (((double) M_PI) / z2)))) - (0.5 * (1.0 / (z2 * ((double) M_PI)))))) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 2e-46) {
		tmp = -Math.atan((((((((Math.PI * Math.PI) * -2.0) * z0) * z0) - -1.0) / (Math.sin(((-2.0 * z0) * Math.PI)) * -z2)) * z1));
	} else {
		tmp = Math.atan(((z1 * ((Math.pow(z0, 2.0) * ((Math.PI / z2) - (0.3333333333333333 * (Math.PI / z2)))) - (0.5 * (1.0 / (z2 * Math.PI))))) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 2e-46:
		tmp = -math.atan((((((((math.pi * math.pi) * -2.0) * z0) * z0) - -1.0) / (math.sin(((-2.0 * z0) * math.pi)) * -z2)) * z1))
	else:
		tmp = math.atan(((z1 * ((math.pow(z0, 2.0) * ((math.pi / z2) - (0.3333333333333333 * (math.pi / z2)))) - (0.5 * (1.0 / (z2 * math.pi))))) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 2e-46)
		tmp = Float64(-atan(Float64(Float64(Float64(Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0) - -1.0) / Float64(sin(Float64(Float64(-2.0 * z0) * pi)) * Float64(-z2))) * z1)));
	else
		tmp = atan(Float64(Float64(z1 * Float64(Float64((z0 ^ 2.0) * Float64(Float64(pi / z2) - Float64(0.3333333333333333 * Float64(pi / z2)))) - Float64(0.5 * Float64(1.0 / Float64(z2 * pi))))) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 2e-46)
		tmp = -atan((((((((pi * pi) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * pi)) * -z2)) * z1));
	else
		tmp = atan(((z1 * (((z0 ^ 2.0) * ((pi / z2) - (0.3333333333333333 * (pi / z2)))) - (0.5 * (1.0 / (z2 * pi))))) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 2e-46], (-N[ArcTan[N[(N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * (-z2)), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]), N[ArcTan[N[(N[(z1 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(Pi / z2), $MachinePrecision] - N[(0.3333333333333333 * N[(Pi / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 2e-46

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(\mathsf{neg}\left(z1\right)\right)} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{neg}\left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      5. atan-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)} \]

   if 2e-46 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z1 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - \frac{1}{3} \cdot \frac{\pi}{z2}\right) - \frac{1}{2} \cdot \frac{1}{z2 \cdot \mathsf{PI}\left(\right)}\right)}{z0}\right) \]

   9. Applied rewrites 67.5%

      \[\leadsto \tan^{-1} \left(\frac{z1 \cdot \left({z0}^{2} \cdot \left(\frac{\pi}{z2} - 0.3333333333333333 \cdot \frac{\pi}{z2}\right) - 0.5 \cdot \frac{1}{z2 \cdot \pi}\right)}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 4 \cdot 10^{-122}:\\ \;\;\;\;-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{-0.5 \cdot \frac{z1}{\pi} + {z0}^{2} \cdot \left(z1 \cdot \pi - 0.3333333333333333 \cdot \left(z1 \cdot \pi\right)\right)}{z2}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 4e-122)
  (-
   (atan
    (*
     (/
      (- (* (* (* (* PI PI) -2.0) z0) z0) -1.0)
      (* (sin (* (* -2.0 z0) PI)) (- z2)))
     z1)))
  (atan
   (/
    (/
     (+
      (* -0.5 (/ z1 PI))
      (* (pow z0 2.0) (- (* z1 PI) (* 0.3333333333333333 (* z1 PI)))))
     z2)
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 4e-122) {
		tmp = -atan((((((((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * ((double) M_PI))) * -z2)) * z1));
	} else {
		tmp = atan(((((-0.5 * (z1 / ((double) M_PI))) + (pow(z0, 2.0) * ((z1 * ((double) M_PI)) - (0.3333333333333333 * (z1 * ((double) M_PI)))))) / z2) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 4e-122) {
		tmp = -Math.atan((((((((Math.PI * Math.PI) * -2.0) * z0) * z0) - -1.0) / (Math.sin(((-2.0 * z0) * Math.PI)) * -z2)) * z1));
	} else {
		tmp = Math.atan(((((-0.5 * (z1 / Math.PI)) + (Math.pow(z0, 2.0) * ((z1 * Math.PI) - (0.3333333333333333 * (z1 * Math.PI))))) / z2) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 4e-122:
		tmp = -math.atan((((((((math.pi * math.pi) * -2.0) * z0) * z0) - -1.0) / (math.sin(((-2.0 * z0) * math.pi)) * -z2)) * z1))
	else:
		tmp = math.atan(((((-0.5 * (z1 / math.pi)) + (math.pow(z0, 2.0) * ((z1 * math.pi) - (0.3333333333333333 * (z1 * math.pi))))) / z2) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 4e-122)
		tmp = Float64(-atan(Float64(Float64(Float64(Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0) - -1.0) / Float64(sin(Float64(Float64(-2.0 * z0) * pi)) * Float64(-z2))) * z1)));
	else
		tmp = atan(Float64(Float64(Float64(Float64(-0.5 * Float64(z1 / pi)) + Float64((z0 ^ 2.0) * Float64(Float64(z1 * pi) - Float64(0.3333333333333333 * Float64(z1 * pi))))) / z2) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 4e-122)
		tmp = -atan((((((((pi * pi) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * pi)) * -z2)) * z1));
	else
		tmp = atan(((((-0.5 * (z1 / pi)) + ((z0 ^ 2.0) * ((z1 * pi) - (0.3333333333333333 * (z1 * pi))))) / z2) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 4e-122], (-N[ArcTan[N[(N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * (-z2)), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]), N[ArcTan[N[(N[(N[(N[(-0.5 * N[(z1 / Pi), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(z1 * Pi), $MachinePrecision] - N[(0.3333333333333333 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 4.0000000000000002e-122

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(\mathsf{neg}\left(z1\right)\right)} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{neg}\left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      5. atan-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)} \]

   if 4.0000000000000002e-122 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]

   7. Taylor expanded in z2 around 0

      \[\leadsto \tan^{-1} \left(\frac{\frac{\frac{-1}{2} \cdot \frac{z1}{\pi} + {z0}^{2} \cdot \left(z1 \cdot \pi - \frac{1}{3} \cdot \left(z1 \cdot \pi\right)\right)}{z2}}{z0}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{\frac{-1}{2} \cdot \frac{z1}{\mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(z1 \cdot \mathsf{PI}\left(\right) - \frac{1}{3} \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)\right)}{z2}}{z0}\right) \]

   9. Applied rewrites 67.4%

      \[\leadsto \tan^{-1} \left(\frac{\frac{-0.5 \cdot \frac{z1}{\pi} + {z0}^{2} \cdot \left(z1 \cdot \pi - 0.3333333333333333 \cdot \left(z1 \cdot \pi\right)\right)}{z2}}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1} \left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)\\ \mathbf{elif}\;z0 \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;-\tan^{-1} \left(\frac{\left(\left(\left(-2 \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \pi - -1}{\left(\left(z0 + z0\right) \cdot z2\right) \cdot \pi} \cdot z1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 -5.6e+15)
  (atan (* z1 (/ (tan (* 0.5 PI)) z2)))
  (if (<= z0 1.6e+112)
    (-
     (atan
      (*
       (/
        (- (* (* (* (* -2.0 z0) z0) PI) PI) -1.0)
        (* (* (+ z0 z0) z2) PI))
       z1)))
    (atan
     (/
      (+
       (* -0.5 (/ z1 (* z2 PI)))
       (/ (* (* 0.6666666666666666 (* z0 z0)) (* PI z1)) z2))
      z0)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= -5.6e+15) {
		tmp = atan((z1 * (tan((0.5 * ((double) M_PI))) / z2)));
	} else if (z0 <= 1.6e+112) {
		tmp = -atan((((((((-2.0 * z0) * z0) * ((double) M_PI)) * ((double) M_PI)) - -1.0) / (((z0 + z0) * z2) * ((double) M_PI))) * z1));
	} else {
		tmp = atan((((-0.5 * (z1 / (z2 * ((double) M_PI)))) + (((0.6666666666666666 * (z0 * z0)) * (((double) M_PI) * z1)) / z2)) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= -5.6e+15) {
		tmp = Math.atan((z1 * (Math.tan((0.5 * Math.PI)) / z2)));
	} else if (z0 <= 1.6e+112) {
		tmp = -Math.atan((((((((-2.0 * z0) * z0) * Math.PI) * Math.PI) - -1.0) / (((z0 + z0) * z2) * Math.PI)) * z1));
	} else {
		tmp = Math.atan((((-0.5 * (z1 / (z2 * Math.PI))) + (((0.6666666666666666 * (z0 * z0)) * (Math.PI * z1)) / z2)) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= -5.6e+15:
		tmp = math.atan((z1 * (math.tan((0.5 * math.pi)) / z2)))
	elif z0 <= 1.6e+112:
		tmp = -math.atan((((((((-2.0 * z0) * z0) * math.pi) * math.pi) - -1.0) / (((z0 + z0) * z2) * math.pi)) * z1))
	else:
		tmp = math.atan((((-0.5 * (z1 / (z2 * math.pi))) + (((0.6666666666666666 * (z0 * z0)) * (math.pi * z1)) / z2)) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= -5.6e+15)
		tmp = atan(Float64(z1 * Float64(tan(Float64(0.5 * pi)) / z2)));
	elseif (z0 <= 1.6e+112)
		tmp = Float64(-atan(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * z0) * z0) * pi) * pi) - -1.0) / Float64(Float64(Float64(z0 + z0) * z2) * pi)) * z1)));
	else
		tmp = atan(Float64(Float64(Float64(-0.5 * Float64(z1 / Float64(z2 * pi))) + Float64(Float64(Float64(0.6666666666666666 * Float64(z0 * z0)) * Float64(pi * z1)) / z2)) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= -5.6e+15)
		tmp = atan((z1 * (tan((0.5 * pi)) / z2)));
	elseif (z0 <= 1.6e+112)
		tmp = -atan((((((((-2.0 * z0) * z0) * pi) * pi) - -1.0) / (((z0 + z0) * z2) * pi)) * z1));
	else
		tmp = atan((((-0.5 * (z1 / (z2 * pi))) + (((0.6666666666666666 * (z0 * z0)) * (pi * z1)) / z2)) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, -5.6e+15], N[ArcTan[N[(z1 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z0, 1.6e+112], (-N[ArcTan[N[(N[(N[(N[(N[(N[(N[(-2.0 * z0), $MachinePrecision] * z0), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(z0 + z0), $MachinePrecision] * z2), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]), N[ArcTan[N[(N[(N[(-0.5 * N[(z1 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -5.6e15

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \color{blue}{0.5}\right)\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \frac{1}{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(\pi \cdot \frac{1}{2}\right)}{z2}\right)} \]

      6. lower-/.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot 0.5\right)}{z2}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}{z2}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}}{z2}\right) \]

      9. lower-*.f64 42.6%

         \[\leadsto \tan^{-1} \left(z1 \cdot \frac{\tan \color{blue}{\left(0.5 \cdot \pi\right)}}{z2}\right) \]

   6. Applied rewrites 42.6%

      \[\leadsto \tan^{-1} \color{blue}{\left(z1 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z2}\right)} \]

   if -5.6e15 < z0 < 1.5999999999999999e112

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   7. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\color{blue}{2 \cdot \left(z0 \cdot \left(z2 \cdot \pi\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{2 \cdot \color{blue}{\left(z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{2 \cdot \left(z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{2 \cdot \left(z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \]

      4. lower-PI.f64 65.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{2 \cdot \left(z0 \cdot \left(z2 \cdot \pi\right)\right)}\right) \]

   9. Applied rewrites 65.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\color{blue}{2 \cdot \left(z0 \cdot \left(z2 \cdot \pi\right)\right)}}\right) \]

   11. Applied rewrites 65.8%

       \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\left(\left(\left(-2 \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \pi - -1}{\left(\left(z0 + z0\right) \cdot z2\right) \cdot \pi} \cdot z1\right)} \]

   if 1.5999999999999999e112 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \frac{\pi \cdot z1}{z2}}{z0}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      12. lower-*.f64 63.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

   10. Applied rewrites 63.2%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 1.75 \cdot 10^{+20}:\\ \;\;\;\;-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 1.75e+20)
  (-
   (atan
    (*
     (/
      (- (* (* (* (* PI PI) -2.0) z0) z0) -1.0)
      (* (sin (* (* -2.0 z0) PI)) (- z2)))
     z1)))
  (atan
   (/
    (+
     (* -0.5 (/ z1 (* z2 PI)))
     (/ (* (* 0.6666666666666666 (* z0 z0)) (* PI z1)) z2))
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.75e+20) {
		tmp = -atan((((((((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * ((double) M_PI))) * -z2)) * z1));
	} else {
		tmp = atan((((-0.5 * (z1 / (z2 * ((double) M_PI)))) + (((0.6666666666666666 * (z0 * z0)) * (((double) M_PI) * z1)) / z2)) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.75e+20) {
		tmp = -Math.atan((((((((Math.PI * Math.PI) * -2.0) * z0) * z0) - -1.0) / (Math.sin(((-2.0 * z0) * Math.PI)) * -z2)) * z1));
	} else {
		tmp = Math.atan((((-0.5 * (z1 / (z2 * Math.PI))) + (((0.6666666666666666 * (z0 * z0)) * (Math.PI * z1)) / z2)) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 1.75e+20:
		tmp = -math.atan((((((((math.pi * math.pi) * -2.0) * z0) * z0) - -1.0) / (math.sin(((-2.0 * z0) * math.pi)) * -z2)) * z1))
	else:
		tmp = math.atan((((-0.5 * (z1 / (z2 * math.pi))) + (((0.6666666666666666 * (z0 * z0)) * (math.pi * z1)) / z2)) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 1.75e+20)
		tmp = Float64(-atan(Float64(Float64(Float64(Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0) - -1.0) / Float64(sin(Float64(Float64(-2.0 * z0) * pi)) * Float64(-z2))) * z1)));
	else
		tmp = atan(Float64(Float64(Float64(-0.5 * Float64(z1 / Float64(z2 * pi))) + Float64(Float64(Float64(0.6666666666666666 * Float64(z0 * z0)) * Float64(pi * z1)) / z2)) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 1.75e+20)
		tmp = -atan((((((((pi * pi) * -2.0) * z0) * z0) - -1.0) / (sin(((-2.0 * z0) * pi)) * -z2)) * z1));
	else
		tmp = atan((((-0.5 * (z1 / (z2 * pi))) + (((0.6666666666666666 * (z0 * z0)) * (pi * z1)) / z2)) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 1.75e+20], (-N[ArcTan[N[(N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * (-z2)), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]), N[ArcTan[N[(N[(N[(-0.5 * N[(z1 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.75e20

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(\mathsf{neg}\left(z1\right)\right)} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{neg}\left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      5. atan-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-\tan^{-1} \left(z1 \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot \left(-z2\right)} \cdot z1\right)} \]

   if 1.75e20 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \frac{\pi \cdot z1}{z2}}{z0}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      12. lower-*.f64 63.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

   10. Applied rewrites 63.2%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 1.75 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1} \left(\frac{z1 \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 1.75e+20)
  (atan
   (/
    (* z1 (- (* (* (* (* PI PI) -2.0) z0) z0) -1.0))
    (* (sin (* (* -2.0 z0) PI)) z2)))
  (atan
   (/
    (+
     (* -0.5 (/ z1 (* z2 PI)))
     (/ (* (* 0.6666666666666666 (* z0 z0)) (* PI z1)) z2))
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.75e+20) {
		tmp = atan(((z1 * (((((((double) M_PI) * ((double) M_PI)) * -2.0) * z0) * z0) - -1.0)) / (sin(((-2.0 * z0) * ((double) M_PI))) * z2)));
	} else {
		tmp = atan((((-0.5 * (z1 / (z2 * ((double) M_PI)))) + (((0.6666666666666666 * (z0 * z0)) * (((double) M_PI) * z1)) / z2)) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 1.75e+20) {
		tmp = Math.atan(((z1 * (((((Math.PI * Math.PI) * -2.0) * z0) * z0) - -1.0)) / (Math.sin(((-2.0 * z0) * Math.PI)) * z2)));
	} else {
		tmp = Math.atan((((-0.5 * (z1 / (z2 * Math.PI))) + (((0.6666666666666666 * (z0 * z0)) * (Math.PI * z1)) / z2)) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 1.75e+20:
		tmp = math.atan(((z1 * (((((math.pi * math.pi) * -2.0) * z0) * z0) - -1.0)) / (math.sin(((-2.0 * z0) * math.pi)) * z2)))
	else:
		tmp = math.atan((((-0.5 * (z1 / (z2 * math.pi))) + (((0.6666666666666666 * (z0 * z0)) * (math.pi * z1)) / z2)) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 1.75e+20)
		tmp = atan(Float64(Float64(z1 * Float64(Float64(Float64(Float64(Float64(pi * pi) * -2.0) * z0) * z0) - -1.0)) / Float64(sin(Float64(Float64(-2.0 * z0) * pi)) * z2)));
	else
		tmp = atan(Float64(Float64(Float64(-0.5 * Float64(z1 / Float64(z2 * pi))) + Float64(Float64(Float64(0.6666666666666666 * Float64(z0 * z0)) * Float64(pi * z1)) / z2)) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 1.75e+20)
		tmp = atan(((z1 * (((((pi * pi) * -2.0) * z0) * z0) - -1.0)) / (sin(((-2.0 * z0) * pi)) * z2)));
	else
		tmp = atan((((-0.5 * (z1 / (z2 * pi))) + (((0.6666666666666666 * (z0 * z0)) * (pi * z1)) / z2)) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 1.75e+20], N[ArcTan[N[(N[(z1 * N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(N[(-2.0 * z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(-0.5 * N[(z1 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.75e20

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + \color{blue}{-2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      6. lower-PI.f64 71.8%

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   6. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\color{blue}{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(-z1\right) \cdot \left(1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(-z1\right) \cdot \left(1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)\right)}{\color{blue}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}}\right) \]

      5. times-frac N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-z1}{-z2} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

      6. lift-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(z1\right)}}{-z2} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\color{blue}{\mathsf{neg}\left(z2\right)}} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

      8. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \frac{1 + -2 \cdot \left({z0}^{2} \cdot {\pi}^{2}\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1 \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot z0\right) \cdot z0 - -1\right)}{\sin \left(\left(-2 \cdot z0\right) \cdot \pi\right) \cdot z2}\right)} \]

   if 1.75e20 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \frac{\pi \cdot z1}{z2}}{z0}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      12. lower-*.f64 63.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

   10. Applied rewrites 63.2%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 8.2 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 8.2e+16)
  (atan (* -0.5 (/ (/ (/ z1 z0) PI) z2)))
  (atan
   (/
    (+
     (* -0.5 (/ z1 (* z2 PI)))
     (/ (* (* 0.6666666666666666 (* z0 z0)) (* PI z1)) z2))
    z0))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 8.2e+16) {
		tmp = atan((-0.5 * (((z1 / z0) / ((double) M_PI)) / z2)));
	} else {
		tmp = atan((((-0.5 * (z1 / (z2 * ((double) M_PI)))) + (((0.6666666666666666 * (z0 * z0)) * (((double) M_PI) * z1)) / z2)) / z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 8.2e+16) {
		tmp = Math.atan((-0.5 * (((z1 / z0) / Math.PI) / z2)));
	} else {
		tmp = Math.atan((((-0.5 * (z1 / (z2 * Math.PI))) + (((0.6666666666666666 * (z0 * z0)) * (Math.PI * z1)) / z2)) / z0));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 8.2e+16:
		tmp = math.atan((-0.5 * (((z1 / z0) / math.pi) / z2)))
	else:
		tmp = math.atan((((-0.5 * (z1 / (z2 * math.pi))) + (((0.6666666666666666 * (z0 * z0)) * (math.pi * z1)) / z2)) / z0))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 8.2e+16)
		tmp = atan(Float64(-0.5 * Float64(Float64(Float64(z1 / z0) / pi) / z2)));
	else
		tmp = atan(Float64(Float64(Float64(-0.5 * Float64(z1 / Float64(z2 * pi))) + Float64(Float64(Float64(0.6666666666666666 * Float64(z0 * z0)) * Float64(pi * z1)) / z2)) / z0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 8.2e+16)
		tmp = atan((-0.5 * (((z1 / z0) / pi) / z2)));
	else
		tmp = atan((((-0.5 * (z1 / (z2 * pi))) + (((0.6666666666666666 * (z0 * z0)) * (pi * z1)) / z2)) / z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 8.2e+16], N[ArcTan[N[(-0.5 * N[(N[(N[(z1 / z0), $MachinePrecision] / Pi), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(-0.5 * N[(z1 / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * N[(z0 * z0), $MachinePrecision]), $MachinePrecision] * N[(Pi * z1), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision] / z0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 8.2e16

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \pi\right)}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2 \cdot \pi}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{z2 \cdot \color{blue}{\pi}}\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\pi \cdot \color{blue}{z2}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right) \]

      9. lower-/.f64 59.0%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

   if 8.2e16 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot \frac{z1}{z2}\right)}{z0}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \frac{\pi \cdot z1}{z2}}{z0}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\left(z0 \cdot z0\right) \cdot \frac{2}{3}\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(\frac{2}{3} \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

      12. lower-*.f64 63.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]

   10. Applied rewrites 63.2%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \frac{\left(0.6666666666666666 \cdot \left(z0 \cdot z0\right)\right) \cdot \left(\pi \cdot z1\right)}{z2}}{z0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 33:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 33.0)
  (atan (* -0.5 (/ (/ (/ z1 z0) PI) z2)))
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = atan((-0.5 * (((z1 / z0) / ((double) M_PI)) / z2)));
	} else {
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = Math.atan((-0.5 * (((z1 / z0) / Math.PI) / z2)));
	} else {
		tmp = Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 33.0:
		tmp = math.atan((-0.5 * (((z1 / z0) / math.pi) / z2)))
	else:
		tmp = math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 33.0)
		tmp = atan(Float64(-0.5 * Float64(Float64(Float64(z1 / z0) / pi) / z2)));
	else
		tmp = atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 33.0)
		tmp = atan((-0.5 * (((z1 / z0) / pi) / z2)));
	else
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 33.0], N[ArcTan[N[(-0.5 * N[(N[(N[(z1 / z0), $MachinePrecision] / Pi), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 33

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \pi\right)}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2 \cdot \pi}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{z2 \cdot \color{blue}{\pi}}\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\pi \cdot \color{blue}{z2}}\right) \]

      6. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right) \]

      9. lower-/.f64 59.0%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{z2}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{\frac{z1}{z0}}{\pi}}{\color{blue}{z2}}\right) \]

   if 33 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   9. Taylor expanded in z0 around inf

      \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       5. lower-PI.f64 21.8%

          \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

   11. Applied rewrites 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 33:\\ \;\;\;\;\tan^{-1} \left(\frac{-0.5}{z2 \cdot z0} \cdot \frac{z1}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 33.0)
  (atan (* (/ -0.5 (* z2 z0)) (/ z1 PI)))
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = atan(((-0.5 / (z2 * z0)) * (z1 / ((double) M_PI))));
	} else {
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = Math.atan(((-0.5 / (z2 * z0)) * (z1 / Math.PI)));
	} else {
		tmp = Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 33.0:
		tmp = math.atan(((-0.5 / (z2 * z0)) * (z1 / math.pi)))
	else:
		tmp = math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 33.0)
		tmp = atan(Float64(Float64(-0.5 / Float64(z2 * z0)) * Float64(z1 / pi)));
	else
		tmp = atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 33.0)
		tmp = atan(((-0.5 / (z2 * z0)) * (z1 / pi)));
	else
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 33.0], N[ArcTan[N[(N[(-0.5 / N[(z2 * z0), $MachinePrecision]), $MachinePrecision] * N[(z1 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 33

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot z1}{\color{blue}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot z1}{z0 \cdot \color{blue}{\left(z2 \cdot \pi\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot z1}{z0 \cdot \left(z2 \cdot \color{blue}{\pi}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot z1}{\left(z0 \cdot z2\right) \cdot \color{blue}{\pi}}\right) \]

      7. times-frac N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2}}{z0 \cdot z2} \cdot \color{blue}{\frac{z1}{\pi}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2}}{z0 \cdot z2} \cdot \color{blue}{\frac{z1}{\pi}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2}}{z0 \cdot z2} \cdot \frac{\color{blue}{z1}}{\pi}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2}}{z2 \cdot z0} \cdot \frac{z1}{\pi}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2}}{z2 \cdot z0} \cdot \frac{z1}{\pi}\right) \]

      12. lower-/.f64 58.9%

          \[\leadsto \tan^{-1} \left(\frac{-0.5}{z2 \cdot z0} \cdot \frac{z1}{\color{blue}{\pi}}\right) \]

   8. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \left(\frac{-0.5}{z2 \cdot z0} \cdot \color{blue}{\frac{z1}{\pi}}\right) \]

   if 33 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   9. Taylor expanded in z0 around inf

      \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       5. lower-PI.f64 21.8%

          \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

   11. Applied rewrites 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 33:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{\frac{z1}{z0}}{z2 \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 33.0)
  (atan (* -0.5 (/ (/ z1 z0) (* z2 PI))))
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = atan((-0.5 * ((z1 / z0) / (z2 * ((double) M_PI)))));
	} else {
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = Math.atan((-0.5 * ((z1 / z0) / (z2 * Math.PI))));
	} else {
		tmp = Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 33.0:
		tmp = math.atan((-0.5 * ((z1 / z0) / (z2 * math.pi))))
	else:
		tmp = math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 33.0)
		tmp = atan(Float64(-0.5 * Float64(Float64(z1 / z0) / Float64(z2 * pi))));
	else
		tmp = atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 33.0)
		tmp = atan((-0.5 * ((z1 / z0) / (z2 * pi))));
	else
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 33.0], N[ArcTan[N[(-0.5 * N[(N[(z1 / z0), $MachinePrecision] / N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 33

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \pi\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \pi\right)}}\right) \]

      3. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2 \cdot \pi}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2 \cdot \pi}}\right) \]

      5. lower-/.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2} \cdot \pi}\right) \]

   8. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{\frac{z1}{z0}}{\color{blue}{z2 \cdot \pi}}\right) \]

   if 33 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   9. Taylor expanded in z0 around inf

      \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       5. lower-PI.f64 21.8%

          \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

   11. Applied rewrites 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 33:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{z1}{\left(z2 \cdot z0\right) \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 33.0)
  (atan (* -0.5 (/ z1 (* (* z2 z0) PI))))
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = atan((-0.5 * (z1 / ((z2 * z0) * ((double) M_PI)))));
	} else {
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = Math.atan((-0.5 * (z1 / ((z2 * z0) * Math.PI))));
	} else {
		tmp = Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 33.0:
		tmp = math.atan((-0.5 * (z1 / ((z2 * z0) * math.pi))))
	else:
		tmp = math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 33.0)
		tmp = atan(Float64(-0.5 * Float64(z1 / Float64(Float64(z2 * z0) * pi))));
	else
		tmp = atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 33.0)
		tmp = atan((-0.5 * (z1 / ((z2 * z0) * pi))));
	else
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 33.0], N[ArcTan[N[(-0.5 * N[(z1 / N[(N[(z2 * z0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 33

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \pi\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\pi}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\left(z0 \cdot z2\right) \cdot \color{blue}{\pi}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\left(z0 \cdot z2\right) \cdot \color{blue}{\pi}}\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\left(z2 \cdot z0\right) \cdot \pi}\right) \]

      6. lower-*.f64 59.0%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{\left(z2 \cdot z0\right) \cdot \pi}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{\left(z2 \cdot z0\right) \cdot \color{blue}{\pi}}\right) \]

   if 33 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   9. Taylor expanded in z0 around inf

      \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       5. lower-PI.f64 21.8%

          \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

   11. Applied rewrites 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq 33:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= z0 33.0)
  (atan (* -0.5 (/ z1 (* z0 (* z2 PI)))))
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2)))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))));
	} else {
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (z0 <= 33.0) {
		tmp = Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))));
	} else {
		tmp = Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if z0 <= 33.0:
		tmp = math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))
	else:
		tmp = math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (z0 <= 33.0)
		tmp = atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))));
	else
		tmp = atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (z0 <= 33.0)
		tmp = atan((-0.5 * (z1 / (z0 * (z2 * pi)))));
	else
		tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[z0, 33.0], N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 33

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z1}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{\color{blue}{z0 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-1}{2} \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lower-PI.f64 58.9%

         \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right) \]

   6. Applied rewrites 58.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)} \]

   if 33 < z0

   1. Initial program 32.3%

      \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

      4. lift-tan.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      5. tan-quot N/A

         \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

      6. frac-times N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   3. Applied rewrites 65.1%

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

   6. Applied rewrites 56.2%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      3. lower-*.f64 56.2%

         \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      10. lower-*.f64 56.2%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

      13. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

      14. associate-/l* N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

      16. lower-*.f64 56.3%

          \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   8. Applied rewrites 56.3%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   9. Taylor expanded in z0 around inf

      \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

       5. lower-PI.f64 21.8%

          \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

   11. Applied rewrites 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 21.8% accurate, 1.8× speedup

Math

\[\tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (atan (* 0.6666666666666666 (/ (* z0 (* z1 PI)) z2))))

C

double code(double z1, double z2, double z0) {
	return atan((0.6666666666666666 * ((z0 * (z1 * ((double) M_PI))) / z2)));
}

Java

public static double code(double z1, double z2, double z0) {
	return Math.atan((0.6666666666666666 * ((z0 * (z1 * Math.PI)) / z2)));
}

Python

def code(z1, z2, z0):
	return math.atan((0.6666666666666666 * ((z0 * (z1 * math.pi)) / z2)))

Julia

function code(z1, z2, z0)
	return atan(Float64(0.6666666666666666 * Float64(Float64(z0 * Float64(z1 * pi)) / z2)))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = atan((0.6666666666666666 * ((z0 * (z1 * pi)) / z2)));
end

Wolfram

code[z1_, z2_, z0_] := N[ArcTan[N[(0.6666666666666666 * N[(N[(z0 * N[(z1 * Pi), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 32.3%

   \[\tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(0.5 + \left(z0 + z0\right)\right)\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \tan^{-1} \left(\color{blue}{\frac{z1}{z2}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

   3. frac-2neg N/A

      \[\leadsto \tan^{-1} \left(\color{blue}{\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)}} \cdot \tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)\right) \]

   4. lift-tan.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

   5. tan-quot N/A

      \[\leadsto \tan^{-1} \left(\frac{\mathsf{neg}\left(z1\right)}{\mathsf{neg}\left(z2\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}}\right) \]

   6. frac-times N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(\mathsf{neg}\left(z1\right)\right) \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \tan^{-1} \color{blue}{\left(\left(\mathsf{neg}\left(z1\right)\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right)} \]

   9. lower-neg.f64 N/A

      \[\leadsto \tan^{-1} \left(\color{blue}{\left(-z1\right)} \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}\right) \]

   10. *-commutative N/A

       \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\color{blue}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \tan^{-1} \left(\left(-z1\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right)}{\cos \left(\pi \cdot \left(\frac{1}{2} + \left(z0 + z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z2\right)\right)}}\right) \]

3. Applied rewrites 65.1%

   \[\leadsto \tan^{-1} \color{blue}{\left(\left(-z1\right) \cdot \frac{\cos \left(\left(z0 + z0\right) \cdot \pi\right)}{\left(-z2\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)}\right)} \]

4. Taylor expanded in z0 around 0

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \mathsf{PI}\left(\right)} + {z0}^{2} \cdot \left(\frac{z1 \cdot \mathsf{PI}\left(\right)}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \mathsf{PI}\left(\right)}{z2}\right)}{\color{blue}{z0}}\right) \]

6. Applied rewrites 56.2%

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right)} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + {z0}^{2} \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   2. unpow2 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   3. lower-*.f64 56.2%

      \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - 0.3333333333333333 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} - \frac{1}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   6. fp-cancel-sub-sign-inv N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{z1 \cdot \pi}{z2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   8. metadata-eval N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\left(\frac{-1}{3} + 1\right) \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   9. metadata-eval N/A

      \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   10. lower-*.f64 56.2%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{z1 \cdot \pi}{z2}\right)}{z0}\right) \]

   13. *-commutative N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \frac{\pi \cdot z1}{z2}\right)}{z0}\right) \]

   14. associate-/l* N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{\frac{-1}{2} \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(\frac{2}{3} \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

   16. lower-*.f64 56.3%

       \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

8. Applied rewrites 56.3%

   \[\leadsto \tan^{-1} \left(\frac{-0.5 \cdot \frac{z1}{z2 \cdot \pi} + \left(z0 \cdot z0\right) \cdot \left(0.6666666666666666 \cdot \left(\pi \cdot \frac{z1}{z2}\right)\right)}{z0}\right) \]

9. Taylor expanded in z0 around inf

   \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{z2}}\right) \]

    2. lower-/.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \tan^{-1} \left(\frac{2}{3} \cdot \frac{z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)}{z2}\right) \]

    5. lower-PI.f64 21.8%

       \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}\right) \]

11. Applied rewrites 21.8%

    \[\leadsto \tan^{-1} \left(0.6666666666666666 \cdot \color{blue}{\frac{z0 \cdot \left(z1 \cdot \pi\right)}{z2}}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z2 z0)
  :name "(atan (* (/ z1 z2) (tan (* PI (+ 1/2 (+ z0 z0))))))"
  :precision binary64
  (atan (* (/ z1 z2) (tan (* PI (+ 0.5 (+ z0 z0)))))))