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

Percentage Accurate: 31.5% → 98.6%

Time: 5.2s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{z2 - -0.5}{z2} \cdot 1\right) \cdot \left(z2 \cdot \pi\right)\\ t_1 := \frac{\pi + \pi}{0}\\ \mathbf{if}\;z2 \leq -2650000000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - t\_1\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(\sin t\_0 \cdot \cos \left(z2 \cdot \pi\right) + \cos t\_0 \cdot \sin \left(z2 \cdot \pi\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z0 \cdot \sin \left(0.5 \cdot \pi\right)}{z1 \cdot \cos \left(0.5 \cdot \pi\right)} + \frac{z0 \cdot \left(z2 \cdot \left(t\_1 + \left(\pi + \pi\right)\right)\right)}{z1}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (* (/ (- z2 -0.5) z2) 1.0) (* z2 PI)))
       (t_1 (/ (+ PI PI) 0.0)))
  (if (<= z2 -2650000000.0)
    (atan
     (-
      (* (* (- (+ PI PI) t_1) (/ z2 z1)) z0)
      (* (/ -1.0 (* 0.0 z1)) z0)))
    (if (<= z2 0.0066)
      (atan
       (/
        (*
         (+
          (* (sin t_0) (cos (* z2 PI)))
          (* (cos t_0) (sin (* z2 PI))))
         z0)
        (* (- (sin (* PI (+ z2 z2)))) z1)))
      (atan
       (+
        (/ (* z0 (sin (* 0.5 PI))) (* z1 (cos (* 0.5 PI))))
        (/ (* z0 (* z2 (+ t_1 (+ PI PI)))) z1)))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (((z2 - -0.5) / z2) * 1.0) * (z2 * ((double) M_PI));
	double t_1 = (((double) M_PI) + ((double) M_PI)) / 0.0;
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - t_1) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = atan(((((sin(t_0) * cos((z2 * ((double) M_PI)))) + (cos(t_0) * sin((z2 * ((double) M_PI))))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1)));
	} else {
		tmp = atan((((z0 * sin((0.5 * ((double) M_PI)))) / (z1 * cos((0.5 * ((double) M_PI))))) + ((z0 * (z2 * (t_1 + (((double) M_PI) + ((double) M_PI))))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (((z2 - -0.5) / z2) * 1.0) * (z2 * Math.PI);
	double t_1 = (Math.PI + Math.PI) / 0.0;
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - t_1) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = Math.atan(((((Math.sin(t_0) * Math.cos((z2 * Math.PI))) + (Math.cos(t_0) * Math.sin((z2 * Math.PI)))) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1)));
	} else {
		tmp = Math.atan((((z0 * Math.sin((0.5 * Math.PI))) / (z1 * Math.cos((0.5 * Math.PI)))) + ((z0 * (z2 * (t_1 + (Math.PI + Math.PI)))) / z1)));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (((z2 - -0.5) / z2) * 1.0) * (z2 * math.pi)
	t_1 = (math.pi + math.pi) / 0.0
	tmp = 0
	if z2 <= -2650000000.0:
		tmp = math.atan((((((math.pi + math.pi) - t_1) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 0.0066:
		tmp = math.atan(((((math.sin(t_0) * math.cos((z2 * math.pi))) + (math.cos(t_0) * math.sin((z2 * math.pi)))) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1)))
	else:
		tmp = math.atan((((z0 * math.sin((0.5 * math.pi))) / (z1 * math.cos((0.5 * math.pi)))) + ((z0 * (z2 * (t_1 + (math.pi + math.pi)))) / z1)))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(Float64(z2 - -0.5) / z2) * 1.0) * Float64(z2 * pi))
	t_1 = Float64(Float64(pi + pi) / 0.0)
	tmp = 0.0
	if (z2 <= -2650000000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - t_1) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan(Float64(Float64(Float64(Float64(sin(t_0) * cos(Float64(z2 * pi))) + Float64(cos(t_0) * sin(Float64(z2 * pi)))) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1)));
	else
		tmp = atan(Float64(Float64(Float64(z0 * sin(Float64(0.5 * pi))) / Float64(z1 * cos(Float64(0.5 * pi)))) + Float64(Float64(z0 * Float64(z2 * Float64(t_1 + Float64(pi + pi)))) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (((z2 - -0.5) / z2) * 1.0) * (z2 * pi);
	t_1 = (pi + pi) / 0.0;
	tmp = 0.0;
	if (z2 <= -2650000000.0)
		tmp = atan((((((pi + pi) - t_1) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan(((((sin(t_0) * cos((z2 * pi))) + (cos(t_0) * sin((z2 * pi)))) * z0) / (-sin((pi * (z2 + z2))) * z1)));
	else
		tmp = atan((((z0 * sin((0.5 * pi))) / (z1 * cos((0.5 * pi)))) + ((z0 * (z2 * (t_1 + (pi + pi)))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[(N[(z2 - -0.5), $MachinePrecision] / z2), $MachinePrecision] * 1.0), $MachinePrecision] * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]}, If[LessEqual[z2, -2650000000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - t$95$1), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 0.0066], N[ArcTan[N[(N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[(z2 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(z2 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(z0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(z1 * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 * N[(z2 * N[(t$95$1 + N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -2.65e9

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -2.65e9 < z2 < 0.0066

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   5. Applied rewrites 52.7%

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

   if 0.0066 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 25.4%

      \[\leadsto \tan^{-1} \left(\frac{z0 \cdot \sin \left(0.5 \cdot \pi\right)}{z1 \cdot \cos \left(0.5 \cdot \pi\right)} + \frac{z0 \cdot \left(z2 \cdot \left(\frac{\pi + \pi}{0} + \left(\pi + \pi\right)\right)\right)}{z1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\pi + \pi}{0}\\ \mathbf{if}\;z2 \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - t\_0\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\tan^{-1} \left(\frac{\sin \left(\pi \cdot \left(\left(-0.5 - z2\right) - z2\right) + \pi\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{z0 \cdot \sin \left(0.5 \cdot \pi\right)}{z1 \cdot \cos \left(0.5 \cdot \pi\right)} + \frac{z0 \cdot \left(z2 \cdot \left(t\_0 + \left(\pi + \pi\right)\right)\right)}{z1}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ (+ PI PI) 0.0)))
  (if (<= z2 -1.6e+16)
    (atan
     (-
      (* (* (- (+ PI PI) t_0) (/ z2 z1)) z0)
      (* (/ -1.0 (* 0.0 z1)) z0)))
    (if (<= z2 0.0066)
      (atan
       (/
        (* (sin (+ (* PI (- (- -0.5 z2) z2)) PI)) z0)
        (* (- (sin (* PI (+ z2 z2)))) z1)))
      (atan
       (+
        (/ (* z0 (sin (* 0.5 PI))) (* z1 (cos (* 0.5 PI))))
        (/ (* z0 (* z2 (+ t_0 (+ PI PI)))) z1)))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (((double) M_PI) + ((double) M_PI)) / 0.0;
	double tmp;
	if (z2 <= -1.6e+16) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - t_0) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = atan(((sin(((((double) M_PI) * ((-0.5 - z2) - z2)) + ((double) M_PI))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1)));
	} else {
		tmp = atan((((z0 * sin((0.5 * ((double) M_PI)))) / (z1 * cos((0.5 * ((double) M_PI))))) + ((z0 * (z2 * (t_0 + (((double) M_PI) + ((double) M_PI))))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (Math.PI + Math.PI) / 0.0;
	double tmp;
	if (z2 <= -1.6e+16) {
		tmp = Math.atan((((((Math.PI + Math.PI) - t_0) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = Math.atan(((Math.sin(((Math.PI * ((-0.5 - z2) - z2)) + Math.PI)) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1)));
	} else {
		tmp = Math.atan((((z0 * Math.sin((0.5 * Math.PI))) / (z1 * Math.cos((0.5 * Math.PI)))) + ((z0 * (z2 * (t_0 + (Math.PI + Math.PI)))) / z1)));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (math.pi + math.pi) / 0.0
	tmp = 0
	if z2 <= -1.6e+16:
		tmp = math.atan((((((math.pi + math.pi) - t_0) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 0.0066:
		tmp = math.atan(((math.sin(((math.pi * ((-0.5 - z2) - z2)) + math.pi)) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1)))
	else:
		tmp = math.atan((((z0 * math.sin((0.5 * math.pi))) / (z1 * math.cos((0.5 * math.pi)))) + ((z0 * (z2 * (t_0 + (math.pi + math.pi)))) / z1)))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(pi + pi) / 0.0)
	tmp = 0.0
	if (z2 <= -1.6e+16)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - t_0) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan(Float64(Float64(sin(Float64(Float64(pi * Float64(Float64(-0.5 - z2) - z2)) + pi)) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1)));
	else
		tmp = atan(Float64(Float64(Float64(z0 * sin(Float64(0.5 * pi))) / Float64(z1 * cos(Float64(0.5 * pi)))) + Float64(Float64(z0 * Float64(z2 * Float64(t_0 + Float64(pi + pi)))) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (pi + pi) / 0.0;
	tmp = 0.0;
	if (z2 <= -1.6e+16)
		tmp = atan((((((pi + pi) - t_0) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan(((sin(((pi * ((-0.5 - z2) - z2)) + pi)) * z0) / (-sin((pi * (z2 + z2))) * z1)));
	else
		tmp = atan((((z0 * sin((0.5 * pi))) / (z1 * cos((0.5 * pi)))) + ((z0 * (z2 * (t_0 + (pi + pi)))) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]}, If[LessEqual[z2, -1.6e+16], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 0.0066], N[ArcTan[N[(N[(N[Sin[N[(N[(Pi * N[(N[(-0.5 - z2), $MachinePrecision] - z2), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(z0 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(z1 * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 * N[(z2 * N[(t$95$0 + N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -1.6e16

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -1.6e16 < z2 < 0.0066

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-PI.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sub-negate-rev N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-+.f64 N/A

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

      17. associate--r+ N/A

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

      18. lift--.f64 N/A

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

      19. distribute-rgt-out-- N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   if 0.0066 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 25.4%

      \[\leadsto \tan^{-1} \left(\frac{z0 \cdot \sin \left(0.5 \cdot \pi\right)}{z1 \cdot \cos \left(0.5 \cdot \pi\right)} + \frac{z0 \cdot \left(z2 \cdot \left(\frac{\pi + \pi}{0} + \left(\pi + \pi\right)\right)\right)}{z1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z2 \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 1.4:\\ \;\;\;\;\tan^{-1} \left(\frac{\sin \left(\pi \cdot \left(\left(-0.5 - z2\right) - z2\right) + \pi\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{z1}{z0} \cdot \tan \left(\left(-0.5 \cdot \pi\right) + \pi \cdot 0.5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= z2 -1.6e+16)
  (atan
   (-
    (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
    (* (/ -1.0 (* 0.0 z1)) z0)))
  (if (<= z2 1.4)
    (atan
     (/
      (* (sin (+ (* PI (- (- -0.5 z2) z2)) PI)) z0)
      (* (- (sin (* PI (+ z2 z2)))) z1)))
    (atan (/ 1.0 (* (/ z1 z0) (tan (+ (- (* 0.5 PI)) (* PI 0.5)))))))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -1.6e+16) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = atan(((sin(((((double) M_PI) * ((-0.5 - z2) - z2)) + ((double) M_PI))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1)));
	} else {
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * ((double) M_PI)) + (((double) M_PI) * 0.5))))));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -1.6e+16) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = Math.atan(((Math.sin(((Math.PI * ((-0.5 - z2) - z2)) + Math.PI)) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1)));
	} else {
		tmp = Math.atan((1.0 / ((z1 / z0) * Math.tan((-(0.5 * Math.PI) + (Math.PI * 0.5))))));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if z2 <= -1.6e+16:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 1.4:
		tmp = math.atan(((math.sin(((math.pi * ((-0.5 - z2) - z2)) + math.pi)) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1)))
	else:
		tmp = math.atan((1.0 / ((z1 / z0) * math.tan((-(0.5 * math.pi) + (math.pi * 0.5))))))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (z2 <= -1.6e+16)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan(Float64(Float64(sin(Float64(Float64(pi * Float64(Float64(-0.5 - z2) - z2)) + pi)) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1)));
	else
		tmp = atan(Float64(1.0 / Float64(Float64(z1 / z0) * tan(Float64(Float64(-Float64(0.5 * pi)) + Float64(pi * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (z2 <= -1.6e+16)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan(((sin(((pi * ((-0.5 - z2) - z2)) + pi)) * z0) / (-sin((pi * (z2 + z2))) * z1)));
	else
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * pi) + (pi * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[z2, -1.6e+16], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 1.4], N[ArcTan[N[(N[(N[Sin[N[(N[(Pi * N[(N[(-0.5 - z2), $MachinePrecision] - z2), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(N[(z1 / z0), $MachinePrecision] * N[Tan[N[((-N[(0.5 * Pi), $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -1.6e16

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -1.6e16 < z2 < 1.3999999999999999

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-PI.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sub-negate-rev N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-+.f64 N/A

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

      17. associate--r+ N/A

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

      18. lift--.f64 N/A

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

      19. distribute-rgt-out-- N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   if 1.3999999999999999 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 43.0%

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

   6. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{z1}{\tan \left(0.5 \cdot \pi\right) \cdot z0}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-+PI/2-rev N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 59.9%

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

   8. Applied rewrites 59.9%

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

Alternative 4: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(z2 + z2\right)\\ \mathbf{if}\;z2 \leq -2650000000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 1.4:\\ \;\;\;\;\tan^{-1} \left(\frac{\cos t\_0 \cdot z0}{\left(-\sin t\_0\right) \cdot z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{z1}{z0} \cdot \tan \left(\left(-0.5 \cdot \pi\right) + \pi \cdot 0.5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* PI (+ z2 z2))))
  (if (<= z2 -2650000000.0)
    (atan
     (-
      (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
      (* (/ -1.0 (* 0.0 z1)) z0)))
    (if (<= z2 1.4)
      (atan (/ (* (cos t_0) z0) (* (- (sin t_0)) z1)))
      (atan
       (/ 1.0 (* (/ z1 z0) (tan (+ (- (* 0.5 PI)) (* PI 0.5))))))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((double) M_PI) * (z2 + z2);
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = atan(((cos(t_0) * z0) / (-sin(t_0) * z1)));
	} else {
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * ((double) M_PI)) + (((double) M_PI) * 0.5))))));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.PI * (z2 + z2);
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = Math.atan(((Math.cos(t_0) * z0) / (-Math.sin(t_0) * z1)));
	} else {
		tmp = Math.atan((1.0 / ((z1 / z0) * Math.tan((-(0.5 * Math.PI) + (Math.PI * 0.5))))));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pi * (z2 + z2)
	tmp = 0
	if z2 <= -2650000000.0:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 1.4:
		tmp = math.atan(((math.cos(t_0) * z0) / (-math.sin(t_0) * z1)))
	else:
		tmp = math.atan((1.0 / ((z1 / z0) * math.tan((-(0.5 * math.pi) + (math.pi * 0.5))))))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(pi * Float64(z2 + z2))
	tmp = 0.0
	if (z2 <= -2650000000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan(Float64(Float64(cos(t_0) * z0) / Float64(Float64(-sin(t_0)) * z1)));
	else
		tmp = atan(Float64(1.0 / Float64(Float64(z1 / z0) * tan(Float64(Float64(-Float64(0.5 * pi)) + Float64(pi * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = pi * (z2 + z2);
	tmp = 0.0;
	if (z2 <= -2650000000.0)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan(((cos(t_0) * z0) / (-sin(t_0) * z1)));
	else
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * pi) + (pi * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z2, -2650000000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 1.4], N[ArcTan[N[(N[(N[Cos[t$95$0], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[t$95$0], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(N[(z1 / z0), $MachinePrecision] * N[Tan[N[((-N[(0.5 * Pi), $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -2.65e9

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -2.65e9 < z2 < 1.3999999999999999

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   if 1.3999999999999999 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 43.0%

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

   6. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{z1}{\tan \left(0.5 \cdot \pi\right) \cdot z0}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-+PI/2-rev N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 59.9%

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

   8. Applied rewrites 59.9%

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

Alternative 5: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(z2 + z2\right)\\ \mathbf{if}\;z2 \leq -2650000000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 1.4:\\ \;\;\;\;\tan^{-1} \left(z0 \cdot \frac{\cos t\_0}{\left(-\sin t\_0\right) \cdot z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{z1}{z0} \cdot \tan \left(\left(-0.5 \cdot \pi\right) + \pi \cdot 0.5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* PI (+ z2 z2))))
  (if (<= z2 -2650000000.0)
    (atan
     (-
      (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
      (* (/ -1.0 (* 0.0 z1)) z0)))
    (if (<= z2 1.4)
      (atan (* z0 (/ (cos t_0) (* (- (sin t_0)) z1))))
      (atan
       (/ 1.0 (* (/ z1 z0) (tan (+ (- (* 0.5 PI)) (* PI 0.5))))))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((double) M_PI) * (z2 + z2);
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = atan((z0 * (cos(t_0) / (-sin(t_0) * z1))));
	} else {
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * ((double) M_PI)) + (((double) M_PI) * 0.5))))));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.PI * (z2 + z2);
	double tmp;
	if (z2 <= -2650000000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 1.4) {
		tmp = Math.atan((z0 * (Math.cos(t_0) / (-Math.sin(t_0) * z1))));
	} else {
		tmp = Math.atan((1.0 / ((z1 / z0) * Math.tan((-(0.5 * Math.PI) + (Math.PI * 0.5))))));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pi * (z2 + z2)
	tmp = 0
	if z2 <= -2650000000.0:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 1.4:
		tmp = math.atan((z0 * (math.cos(t_0) / (-math.sin(t_0) * z1))))
	else:
		tmp = math.atan((1.0 / ((z1 / z0) * math.tan((-(0.5 * math.pi) + (math.pi * 0.5))))))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(pi * Float64(z2 + z2))
	tmp = 0.0
	if (z2 <= -2650000000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan(Float64(z0 * Float64(cos(t_0) / Float64(Float64(-sin(t_0)) * z1))));
	else
		tmp = atan(Float64(1.0 / Float64(Float64(z1 / z0) * tan(Float64(Float64(-Float64(0.5 * pi)) + Float64(pi * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = pi * (z2 + z2);
	tmp = 0.0;
	if (z2 <= -2650000000.0)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 1.4)
		tmp = atan((z0 * (cos(t_0) / (-sin(t_0) * z1))));
	else
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * pi) + (pi * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z2, -2650000000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 1.4], N[ArcTan[N[(z0 * N[(N[Cos[t$95$0], $MachinePrecision] / N[((-N[Sin[t$95$0], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(N[(z1 / z0), $MachinePrecision] * N[Tan[N[((-N[(0.5 * Pi), $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -2.65e9

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -2.65e9 < z2 < 1.3999999999999999

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. frac-times N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   if 1.3999999999999999 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 43.0%

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

   6. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{z1}{\tan \left(0.5 \cdot \pi\right) \cdot z0}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-+PI/2-rev N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 59.9%

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

   8. Applied rewrites 59.9%

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

Alternative 6: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z2 \leq -4500000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{z0}{\left(z1 \cdot \pi\right) \cdot z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{z1}{z0} \cdot \tan \left(\left(-0.5 \cdot \pi\right) + \pi \cdot 0.5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= z2 -4500000.0)
  (atan
   (-
    (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
    (* (/ -1.0 (* 0.0 z1)) z0)))
  (if (<= z2 0.0066)
    (atan (* -0.5 (/ z0 (* (* z1 PI) z2))))
    (atan (/ 1.0 (* (/ z1 z0) (tan (+ (- (* 0.5 PI)) (* PI 0.5)))))))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = atan((-0.5 * (z0 / ((z1 * ((double) M_PI)) * z2))));
	} else {
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * ((double) M_PI)) + (((double) M_PI) * 0.5))))));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 0.0066) {
		tmp = Math.atan((-0.5 * (z0 / ((z1 * Math.PI) * z2))));
	} else {
		tmp = Math.atan((1.0 / ((z1 / z0) * Math.tan((-(0.5 * Math.PI) + (Math.PI * 0.5))))));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if z2 <= -4500000.0:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 0.0066:
		tmp = math.atan((-0.5 * (z0 / ((z1 * math.pi) * z2))))
	else:
		tmp = math.atan((1.0 / ((z1 / z0) * math.tan((-(0.5 * math.pi) + (math.pi * 0.5))))))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (z2 <= -4500000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan(Float64(-0.5 * Float64(z0 / Float64(Float64(z1 * pi) * z2))));
	else
		tmp = atan(Float64(1.0 / Float64(Float64(z1 / z0) * tan(Float64(Float64(-Float64(0.5 * pi)) + Float64(pi * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (z2 <= -4500000.0)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 0.0066)
		tmp = atan((-0.5 * (z0 / ((z1 * pi) * z2))));
	else
		tmp = atan((1.0 / ((z1 / z0) * tan((-(0.5 * pi) + (pi * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[z2, -4500000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 0.0066], N[ArcTan[N[(-0.5 * N[(z0 / N[(N[(z1 * Pi), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 / N[(N[(z1 / z0), $MachinePrecision] * N[Tan[N[((-N[(0.5 * Pi), $MachinePrecision]) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -4.5e6

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -4.5e6 < z2 < 0.0066

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   4. Taylor expanded in z2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 58.7%

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

   8. Applied rewrites 58.7%

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

   if 0.0066 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lower-*.f64 43.0%

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

   6. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{z1}{\tan \left(0.5 \cdot \pi\right) \cdot z0}}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-+PI/2-rev N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 59.9%

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

   8. Applied rewrites 59.9%

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

Alternative 7: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z2 \leq -4500000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{elif}\;z2 \leq 11500000:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{z0}{\left(z1 \cdot \pi\right) \cdot z2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(z0 \cdot \frac{\tan \left(0.5 \cdot \pi\right)}{z1}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= z2 -4500000.0)
  (atan
   (-
    (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
    (* (/ -1.0 (* 0.0 z1)) z0)))
  (if (<= z2 11500000.0)
    (atan (* -0.5 (/ z0 (* (* z1 PI) z2))))
    (atan (* z0 (/ (tan (* 0.5 PI)) z1))))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 11500000.0) {
		tmp = atan((-0.5 * (z0 / ((z1 * ((double) M_PI)) * z2))));
	} else {
		tmp = atan((z0 * (tan((0.5 * ((double) M_PI))) / z1)));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else if (z2 <= 11500000.0) {
		tmp = Math.atan((-0.5 * (z0 / ((z1 * Math.PI) * z2))));
	} else {
		tmp = Math.atan((z0 * (Math.tan((0.5 * Math.PI)) / z1)));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if z2 <= -4500000.0:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	elif z2 <= 11500000.0:
		tmp = math.atan((-0.5 * (z0 / ((z1 * math.pi) * z2))))
	else:
		tmp = math.atan((z0 * (math.tan((0.5 * math.pi)) / z1)))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (z2 <= -4500000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	elseif (z2 <= 11500000.0)
		tmp = atan(Float64(-0.5 * Float64(z0 / Float64(Float64(z1 * pi) * z2))));
	else
		tmp = atan(Float64(z0 * Float64(tan(Float64(0.5 * pi)) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (z2 <= -4500000.0)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	elseif (z2 <= 11500000.0)
		tmp = atan((-0.5 * (z0 / ((z1 * pi) * z2))));
	else
		tmp = atan((z0 * (tan((0.5 * pi)) / z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[z2, -4500000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 11500000.0], N[ArcTan[N[(-0.5 * N[(z0 / N[(N[(z1 * Pi), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(z0 * N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -4.5e6

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -4.5e6 < z2 < 1.15e7

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   4. Taylor expanded in z2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 58.7%

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

   8. Applied rewrites 58.7%

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

   if 1.15e7 < z2

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 43.0%

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

   6. Applied rewrites 43.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\tan \left(0.5 \cdot \pi\right) \cdot z0}{z1}\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.0%

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

   8. Applied rewrites 43.0%

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

Alternative 8: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z2 \leq -4500000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \frac{-1}{0 \cdot z1} \cdot z0\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(-0.5 \cdot \frac{z0}{\left(z1 \cdot \pi\right) \cdot z2}\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= z2 -4500000.0)
  (atan
   (-
    (* (* (- (+ PI PI) (/ (+ PI PI) 0.0)) (/ z2 z1)) z0)
    (* (/ -1.0 (* 0.0 z1)) z0)))
  (atan (* -0.5 (/ z0 (* (* z1 PI) z2))))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = atan((((((((double) M_PI) + ((double) M_PI)) - ((((double) M_PI) + ((double) M_PI)) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else {
		tmp = atan((-0.5 * (z0 / ((z1 * ((double) M_PI)) * z2))));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (z2 <= -4500000.0) {
		tmp = Math.atan((((((Math.PI + Math.PI) - ((Math.PI + Math.PI) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	} else {
		tmp = Math.atan((-0.5 * (z0 / ((z1 * Math.PI) * z2))));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if z2 <= -4500000.0:
		tmp = math.atan((((((math.pi + math.pi) - ((math.pi + math.pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)))
	else:
		tmp = math.atan((-0.5 * (z0 / ((z1 * math.pi) * z2))))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (z2 <= -4500000.0)
		tmp = atan(Float64(Float64(Float64(Float64(Float64(pi + pi) - Float64(Float64(pi + pi) / 0.0)) * Float64(z2 / z1)) * z0) - Float64(Float64(-1.0 / Float64(0.0 * z1)) * z0)));
	else
		tmp = atan(Float64(-0.5 * Float64(z0 / Float64(Float64(z1 * pi) * z2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (z2 <= -4500000.0)
		tmp = atan((((((pi + pi) - ((pi + pi) / 0.0)) * (z2 / z1)) * z0) - ((-1.0 / (0.0 * z1)) * z0)));
	else
		tmp = atan((-0.5 * (z0 / ((z1 * pi) * z2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[z2, -4500000.0], N[ArcTan[N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] - N[(N[(Pi + Pi), $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[(N[(-1.0 / N[(0.0 * z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(-0.5 * N[(z0 / N[(N[(z1 * Pi), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z2 < -4.5e6

   1. Initial program 31.5%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 38.8%

      \[\leadsto \tan^{-1} \left(\left(\left(\left(\pi + \pi\right) - \frac{\pi + \pi}{0}\right) \cdot \frac{z2}{z1}\right) \cdot z0 - \color{blue}{\frac{-1}{0 \cdot z1} \cdot z0}\right) \]

   if -4.5e6 < z2

   1. Initial program 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 64.9%

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

   4. Taylor expanded in z2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 58.7%

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

   8. Applied rewrites 58.7%

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

Alternative 9: 58.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 64.9%

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

4. Taylor expanded in z2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 58.7%

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

6. Applied rewrites 58.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 58.7%

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

8. Applied rewrites 58.7%

   \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z0}{\left(z1 \cdot \pi\right) \cdot \color{blue}{z2}}\right) \]
9. Add Preprocessing

Alternative 10: 58.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 64.9%

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

4. Taylor expanded in z2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 58.7%

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

6. Applied rewrites 58.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 58.7%

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

8. Applied rewrites 58.7%

   \[\leadsto \tan^{-1} \left(-0.5 \cdot \frac{z0}{\left(z1 \cdot z2\right) \cdot \color{blue}{\pi}}\right) \]
9. Add Preprocessing

Alternative 11: 58.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 64.9%

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

4. Taylor expanded in z2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 58.7%

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

6. Applied rewrites 58.7%

   \[\leadsto \tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z2 z0 z1)
  :name "(atan (* (tan (* (- (+ z2 z2) -1/2) PI)) (/ z0 z1)))"
  :precision binary64
  (atan (* (tan (* (- (+ z2 z2) -0.5) PI)) (/ z0 z1))))