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

Percentage Accurate: 56.7% → 87.0%

Time: 8.3s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (/
 (-
  1.0
  (cos (* -2.0 (atan (* (/ z1 z2) (tan (* PI (- (+ z0 z0) -0.5))))))))
 (* (+ z1 z1) z1)))

C

double code(double z1, double z2, double z0) {
	return (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((((double) M_PI) * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
}

Java

public static double code(double z1, double z2, double z0) {
	return (1.0 - Math.cos((-2.0 * Math.atan(((z1 / z2) * Math.tan((Math.PI * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
}

Python

def code(z1, z2, z0):
	return (1.0 - math.cos((-2.0 * math.atan(((z1 / z2) * math.tan((math.pi * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1)

Julia

function code(z1, z2, z0)
	return Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(Float64(z1 / z2) * tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5)))))))) / Float64(Float64(z1 + z1) * z1))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((pi * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
end

Wolfram

code[z1_, z2_, z0_] := N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]

Initial Program: 56.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (/
 (-
  1.0
  (cos (* -2.0 (atan (* (/ z1 z2) (tan (* PI (- (+ z0 z0) -0.5))))))))
 (* (+ z1 z1) z1)))

C

double code(double z1, double z2, double z0) {
	return (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((((double) M_PI) * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
}

Java

public static double code(double z1, double z2, double z0) {
	return (1.0 - Math.cos((-2.0 * Math.atan(((z1 / z2) * Math.tan((Math.PI * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
}

Python

def code(z1, z2, z0):
	return (1.0 - math.cos((-2.0 * math.atan(((z1 / z2) * math.tan((math.pi * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1)

Julia

function code(z1, z2, z0)
	return Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(Float64(z1 / z2) * tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5)))))))) / Float64(Float64(z1 + z1) * z1))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((pi * ((z0 + z0) - -0.5)))))))) / ((z1 + z1) * z1);
end

Wolfram

code[z1_, z2_, z0_] := N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|z1\right| + \left|z1\right|\\ \mathbf{if}\;\left|z1\right| \leq 2.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{0.5}{\left|z1\right| \cdot \left|z1\right|} \cdot \left|z1\right| - \frac{\cos \left(2 \cdot \tan^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{\left|z1\right|}{z2}\right)\right)}{t\_0}}{\left|z1\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(\frac{\left(1 + {z0}^{2} \cdot \left(-2 \cdot {\pi}^{2} + 0.6666666666666666 \cdot \left({z0}^{2} \cdot {\pi}^{4}\right)\right)\right) \cdot \left|z1\right|}{\left(-\sin \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot z2}\right)\right)}{t\_0 \cdot \left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z1) (fabs z1))))
  (if (<= (fabs z1) 2.4e-157)
    (/
     (-
      (* (/ 0.5 (* (fabs z1) (fabs z1))) (fabs z1))
      (/
       (cos (* 2.0 (atan (* (tan (* 0.5 PI)) (/ (fabs z1) z2)))))
       t_0))
     (fabs z1))
    (/
     (-
      1.0
      (cos
       (*
        -2.0
        (atan
         (/
          (*
           (+
            1.0
            (*
             (pow z0 2.0)
             (+
              (* -2.0 (pow PI 2.0))
              (* 0.6666666666666666 (* (pow z0 2.0) (pow PI 4.0))))))
           (fabs z1))
          (* (- (sin (* (+ z0 z0) PI))) z2))))))
     (* t_0 (fabs z1))))))

C

double code(double z1, double z2, double z0) {
	double t_0 = fabs(z1) + fabs(z1);
	double tmp;
	if (fabs(z1) <= 2.4e-157) {
		tmp = (((0.5 / (fabs(z1) * fabs(z1))) * fabs(z1)) - (cos((2.0 * atan((tan((0.5 * ((double) M_PI))) * (fabs(z1) / z2))))) / t_0)) / fabs(z1);
	} else {
		tmp = (1.0 - cos((-2.0 * atan((((1.0 + (pow(z0, 2.0) * ((-2.0 * pow(((double) M_PI), 2.0)) + (0.6666666666666666 * (pow(z0, 2.0) * pow(((double) M_PI), 4.0)))))) * fabs(z1)) / (-sin(((z0 + z0) * ((double) M_PI))) * z2)))))) / (t_0 * fabs(z1));
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = Math.abs(z1) + Math.abs(z1);
	double tmp;
	if (Math.abs(z1) <= 2.4e-157) {
		tmp = (((0.5 / (Math.abs(z1) * Math.abs(z1))) * Math.abs(z1)) - (Math.cos((2.0 * Math.atan((Math.tan((0.5 * Math.PI)) * (Math.abs(z1) / z2))))) / t_0)) / Math.abs(z1);
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((((1.0 + (Math.pow(z0, 2.0) * ((-2.0 * Math.pow(Math.PI, 2.0)) + (0.6666666666666666 * (Math.pow(z0, 2.0) * Math.pow(Math.PI, 4.0)))))) * Math.abs(z1)) / (-Math.sin(((z0 + z0) * Math.PI)) * z2)))))) / (t_0 * Math.abs(z1));
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = math.fabs(z1) + math.fabs(z1)
	tmp = 0
	if math.fabs(z1) <= 2.4e-157:
		tmp = (((0.5 / (math.fabs(z1) * math.fabs(z1))) * math.fabs(z1)) - (math.cos((2.0 * math.atan((math.tan((0.5 * math.pi)) * (math.fabs(z1) / z2))))) / t_0)) / math.fabs(z1)
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((((1.0 + (math.pow(z0, 2.0) * ((-2.0 * math.pow(math.pi, 2.0)) + (0.6666666666666666 * (math.pow(z0, 2.0) * math.pow(math.pi, 4.0)))))) * math.fabs(z1)) / (-math.sin(((z0 + z0) * math.pi)) * z2)))))) / (t_0 * math.fabs(z1))
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(abs(z1) + abs(z1))
	tmp = 0.0
	if (abs(z1) <= 2.4e-157)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(abs(z1) * abs(z1))) * abs(z1)) - Float64(cos(Float64(2.0 * atan(Float64(tan(Float64(0.5 * pi)) * Float64(abs(z1) / z2))))) / t_0)) / abs(z1));
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(Float64(Float64(1.0 + Float64((z0 ^ 2.0) * Float64(Float64(-2.0 * (pi ^ 2.0)) + Float64(0.6666666666666666 * Float64((z0 ^ 2.0) * (pi ^ 4.0)))))) * abs(z1)) / Float64(Float64(-sin(Float64(Float64(z0 + z0) * pi))) * z2)))))) / Float64(t_0 * abs(z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = abs(z1) + abs(z1);
	tmp = 0.0;
	if (abs(z1) <= 2.4e-157)
		tmp = (((0.5 / (abs(z1) * abs(z1))) * abs(z1)) - (cos((2.0 * atan((tan((0.5 * pi)) * (abs(z1) / z2))))) / t_0)) / abs(z1);
	else
		tmp = (1.0 - cos((-2.0 * atan((((1.0 + ((z0 ^ 2.0) * ((-2.0 * (pi ^ 2.0)) + (0.6666666666666666 * ((z0 ^ 2.0) * (pi ^ 4.0)))))) * abs(z1)) / (-sin(((z0 + z0) * pi)) * z2)))))) / (t_0 * abs(z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[Abs[z1], $MachinePrecision] + N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z1], $MachinePrecision], 2.4e-157], N[(N[(N[(N[(0.5 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[z1], $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(N[(N[(1.0 + N[(N[Power[z0, 2.0], $MachinePrecision] * N[(N[(-2.0 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(N[Power[z0, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[((-N[Sin[N[(N[(z0 + z0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]) * z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z1 < 2.4e-157

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.0%

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

   6. Applied rewrites 71.0%

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

   if 2.4e-157 < z1

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-PI.f64 83.8%

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

   6. Applied rewrites 83.8%

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

Alternative 2: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 + z0\right) \cdot \pi\\ \mathbf{if}\;\tan \left(\pi \cdot \left(\left(z0 + z0\right) - -0.5\right)\right) \leq 200:\\ \;\;\;\;\frac{\frac{0.5}{z1 \cdot z1} \cdot z1 - \frac{\cos \left(2 \cdot \tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\infty \cdot \infty\right) \cdot \pi\right)\right) \cdot z0 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z1}{z2}\right)\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5}{z1}}{z1} \cdot z1 - \frac{\cos \left(2 \cdot \left(-\tan^{-1} \left(\frac{\cos t\_0 \cdot z1}{\sin t\_0 \cdot z2}\right)\right)\right)}{z1 + z1}}{z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (+ z0 z0) PI)))
  (if (<= (tan (* PI (- (+ z0 z0) -0.5))) 200.0)
    (/
     (-
      (* (/ 0.5 (* z1 z1)) z1)
      (/
       (cos
        (*
         2.0
         (atan
          (/
           (*
            (-
             (* (* 2.0 (+ PI (* (* INFINITY INFINITY) PI))) z0)
             (tan (* PI -0.5)))
            z1)
           z2))))
       (+ z1 z1)))
     z1)
    (/
     (-
      (* (/ (/ 0.5 z1) z1) z1)
      (/
       (cos (* 2.0 (- (atan (/ (* (cos t_0) z1) (* (sin t_0) z2))))))
       (+ z1 z1)))
     z1))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) * ((double) M_PI);
	double tmp;
	if (tan((((double) M_PI) * ((z0 + z0) - -0.5))) <= 200.0) {
		tmp = (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((((((2.0 * (((double) M_PI) + ((((double) INFINITY) * ((double) INFINITY)) * ((double) M_PI)))) * z0) - tan((((double) M_PI) * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
	} else {
		tmp = ((((0.5 / z1) / z1) * z1) - (cos((2.0 * -atan(((cos(t_0) * z1) / (sin(t_0) * z2))))) / (z1 + z1))) / z1;
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) * Math.PI;
	double tmp;
	if (Math.tan((Math.PI * ((z0 + z0) - -0.5))) <= 200.0) {
		tmp = (((0.5 / (z1 * z1)) * z1) - (Math.cos((2.0 * Math.atan((((((2.0 * (Math.PI + ((Double.POSITIVE_INFINITY * Double.POSITIVE_INFINITY) * Math.PI))) * z0) - Math.tan((Math.PI * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
	} else {
		tmp = ((((0.5 / z1) / z1) * z1) - (Math.cos((2.0 * -Math.atan(((Math.cos(t_0) * z1) / (Math.sin(t_0) * z2))))) / (z1 + z1))) / z1;
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (z0 + z0) * math.pi
	tmp = 0
	if math.tan((math.pi * ((z0 + z0) - -0.5))) <= 200.0:
		tmp = (((0.5 / (z1 * z1)) * z1) - (math.cos((2.0 * math.atan((((((2.0 * (math.pi + ((math.inf * math.inf) * math.pi))) * z0) - math.tan((math.pi * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1
	else:
		tmp = ((((0.5 / z1) / z1) * z1) - (math.cos((2.0 * -math.atan(((math.cos(t_0) * z1) / (math.sin(t_0) * z2))))) / (z1 + z1))) / z1
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(z0 + z0) * pi)
	tmp = 0.0
	if (tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5))) <= 200.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(z1 * z1)) * z1) - Float64(cos(Float64(2.0 * atan(Float64(Float64(Float64(Float64(Float64(2.0 * Float64(pi + Float64(Float64(Inf * Inf) * pi))) * z0) - tan(Float64(pi * -0.5))) * z1) / z2)))) / Float64(z1 + z1))) / z1);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / z1) / z1) * z1) - Float64(cos(Float64(2.0 * Float64(-atan(Float64(Float64(cos(t_0) * z1) / Float64(sin(t_0) * z2)))))) / Float64(z1 + z1))) / z1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (z0 + z0) * pi;
	tmp = 0.0;
	if (tan((pi * ((z0 + z0) - -0.5))) <= 200.0)
		tmp = (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((((((2.0 * (pi + ((Inf * Inf) * pi))) * z0) - tan((pi * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
	else
		tmp = ((((0.5 / z1) / z1) * z1) - (cos((2.0 * -atan(((cos(t_0) * z1) / (sin(t_0) * z2))))) / (z1 + z1))) / z1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(z0 + z0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 200.0], N[(N[(N[(N[(0.5 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[(N[(N[(N[(2.0 * N[(Pi + N[(N[(Infinity * Infinity), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] / z2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / z1), $MachinePrecision] / z1), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * (-N[ArcTan[N[(N[(N[Cos[t$95$0], $MachinePrecision] * z1), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 200

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-+.f64 N/A

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

   6. Applied rewrites 76.4%

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

   8. Applied rewrites 77.1%

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

   9. Evaluated real constant 85.6%

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

   10. Evaluated real constant 85.6%

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

   if 200 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. count-2 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. count-2 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 70.9%

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

   5. Applied rewrites 70.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-*.f64 N/A

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

   7. Applied rewrites 76.3%

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

Alternative 3: 85.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (/
 (-
  (* (/ 0.5 (* z1 z1)) z1)
  (/
   (cos
    (*
     2.0
     (atan
      (/
       (*
        (-
         (* (* 2.0 (+ PI (* (* INFINITY INFINITY) PI))) z0)
         (tan (* PI -0.5)))
        z1)
       z2))))
   (+ z1 z1)))
 z1))

C

double code(double z1, double z2, double z0) {
	return (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((((((2.0 * (((double) M_PI) + ((((double) INFINITY) * ((double) INFINITY)) * ((double) M_PI)))) * z0) - tan((((double) M_PI) * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
}

Java

public static double code(double z1, double z2, double z0) {
	return (((0.5 / (z1 * z1)) * z1) - (Math.cos((2.0 * Math.atan((((((2.0 * (Math.PI + ((Double.POSITIVE_INFINITY * Double.POSITIVE_INFINITY) * Math.PI))) * z0) - Math.tan((Math.PI * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
}

Python

def code(z1, z2, z0):
	return (((0.5 / (z1 * z1)) * z1) - (math.cos((2.0 * math.atan((((((2.0 * (math.pi + ((math.inf * math.inf) * math.pi))) * z0) - math.tan((math.pi * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1

Julia

function code(z1, z2, z0)
	return Float64(Float64(Float64(Float64(0.5 / Float64(z1 * z1)) * z1) - Float64(cos(Float64(2.0 * atan(Float64(Float64(Float64(Float64(Float64(2.0 * Float64(pi + Float64(Float64(Inf * Inf) * pi))) * z0) - tan(Float64(pi * -0.5))) * z1) / z2)))) / Float64(z1 + z1))) / z1)
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((((((2.0 * (pi + ((Inf * Inf) * pi))) * z0) - tan((pi * -0.5))) * z1) / z2)))) / (z1 + z1))) / z1;
end

Wolfram

code[z1_, z2_, z0_] := N[(N[(N[(N[(0.5 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[(N[(N[(N[(2.0 * N[(Pi + N[(N[(Infinity * Infinity), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] / z2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. sub-to-fraction N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 70.3%

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

4. Taylor expanded in z0 around 0

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

   1. lower-+.f64 N/A

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

6. Applied rewrites 76.4%

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

8. Applied rewrites 77.1%

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

9. Evaluated real constant 85.6%

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

10. Evaluated real constant 85.6%

    \[\leadsto \frac{\frac{0.5}{z1 \cdot z1} \cdot z1 - \frac{\cos \left(2 \cdot \tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\infty \cdot \infty\right) \cdot \pi\right)\right) \cdot z0 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z1}{z2}\right)\right)}{z1 + z1}}{z1} \]
11. Add Preprocessing

Alternative 4: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 + z0\right) - -0.5\\ \mathbf{if}\;\tan \left(\pi \cdot t\_0\right) \leq 20000000000:\\ \;\;\;\;\frac{\left(\frac{1}{z1} \cdot \frac{0.5}{z1}\right) \cdot z1 - \frac{\cos \left(2 \cdot \tan^{-1} \left(\tan \left(t\_0 \cdot \pi\right) \cdot \frac{z1}{z2}\right)\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{\left(z1 + z1\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (- (+ z0 z0) -0.5)))
  (if (<= (tan (* PI t_0)) 20000000000.0)
    (/
     (-
      (* (* (/ 1.0 z1) (/ 0.5 z1)) z1)
      (/
       (cos (* 2.0 (atan (* (tan (* t_0 PI)) (/ z1 z2)))))
       (+ z1 z1)))
     z1)
    (/
     (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
     (* (+ z1 z1) z1)))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (tan((((double) M_PI) * t_0)) <= 20000000000.0) {
		tmp = ((((1.0 / z1) * (0.5 / z1)) * z1) - (cos((2.0 * atan((tan((t_0 * ((double) M_PI))) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (Math.tan((Math.PI * t_0)) <= 20000000000.0) {
		tmp = ((((1.0 / z1) * (0.5 / z1)) * z1) - (Math.cos((2.0 * Math.atan((Math.tan((t_0 * Math.PI)) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (z0 + z0) - -0.5
	tmp = 0
	if math.tan((math.pi * t_0)) <= 20000000000.0:
		tmp = ((((1.0 / z1) * (0.5 / z1)) * z1) - (math.cos((2.0 * math.atan((math.tan((t_0 * math.pi)) * (z1 / z2))))) / (z1 + z1))) / z1
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(z0 + z0) - -0.5)
	tmp = 0.0
	if (tan(Float64(pi * t_0)) <= 20000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / z1) * Float64(0.5 / z1)) * z1) - Float64(cos(Float64(2.0 * atan(Float64(tan(Float64(t_0 * pi)) * Float64(z1 / z2))))) / Float64(z1 + z1))) / z1);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (z0 + z0) - -0.5;
	tmp = 0.0;
	if (tan((pi * t_0)) <= 20000000000.0)
		tmp = ((((1.0 / z1) * (0.5 / z1)) * z1) - (cos((2.0 * atan((tan((t_0 * pi)) * (z1 / z2))))) / (z1 + z1))) / z1;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision], 20000000000.0], N[(N[(N[(N[(N[(1.0 / z1), $MachinePrecision] * N[(0.5 / z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[Tan[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 2e10

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. count-2 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. unpow-prod-down N/A

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

      14. lower-unsound-pow.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. inv-pow N/A

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

      17. lower-unsound-pow.f32 N/A

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

      18. lower-pow.f32 N/A

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

      19. inv-pow N/A

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

      20. lower-unsound-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. count-2 N/A

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

      24. associate-/r* N/A

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

      25. metadata-eval N/A

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

      26. lower-/.f64 70.7%

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

   5. Applied rewrites 70.7%

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

   if 2e10 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 + z0\right) - -0.5\\ \mathbf{if}\;\tan \left(\pi \cdot t\_0\right) \leq 20000000000:\\ \;\;\;\;\frac{\frac{\frac{0.5}{z1}}{z1} \cdot z1 - \frac{\cos \left(2 \cdot \tan^{-1} \left(\tan \left(t\_0 \cdot \pi\right) \cdot \frac{z1}{z2}\right)\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{\left(z1 + z1\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (- (+ z0 z0) -0.5)))
  (if (<= (tan (* PI t_0)) 20000000000.0)
    (/
     (-
      (* (/ (/ 0.5 z1) z1) z1)
      (/
       (cos (* 2.0 (atan (* (tan (* t_0 PI)) (/ z1 z2)))))
       (+ z1 z1)))
     z1)
    (/
     (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
     (* (+ z1 z1) z1)))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (tan((((double) M_PI) * t_0)) <= 20000000000.0) {
		tmp = ((((0.5 / z1) / z1) * z1) - (cos((2.0 * atan((tan((t_0 * ((double) M_PI))) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (Math.tan((Math.PI * t_0)) <= 20000000000.0) {
		tmp = ((((0.5 / z1) / z1) * z1) - (Math.cos((2.0 * Math.atan((Math.tan((t_0 * Math.PI)) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (z0 + z0) - -0.5
	tmp = 0
	if math.tan((math.pi * t_0)) <= 20000000000.0:
		tmp = ((((0.5 / z1) / z1) * z1) - (math.cos((2.0 * math.atan((math.tan((t_0 * math.pi)) * (z1 / z2))))) / (z1 + z1))) / z1
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(z0 + z0) - -0.5)
	tmp = 0.0
	if (tan(Float64(pi * t_0)) <= 20000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / z1) / z1) * z1) - Float64(cos(Float64(2.0 * atan(Float64(tan(Float64(t_0 * pi)) * Float64(z1 / z2))))) / Float64(z1 + z1))) / z1);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (z0 + z0) - -0.5;
	tmp = 0.0;
	if (tan((pi * t_0)) <= 20000000000.0)
		tmp = ((((0.5 / z1) / z1) * z1) - (cos((2.0 * atan((tan((t_0 * pi)) * (z1 / z2))))) / (z1 + z1))) / z1;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision], 20000000000.0], N[(N[(N[(N[(N[(0.5 / z1), $MachinePrecision] / z1), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[Tan[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 2e10

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. count-2 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. count-2 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 70.9%

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

   5. Applied rewrites 70.9%

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

   if 2e10 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 6: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan \left(\pi \cdot \left(\left(z0 + z0\right) - -0.5\right)\right) \leq 20000000000:\\ \;\;\;\;\frac{\frac{0.5}{z1 \cdot z1} \cdot z1 - \frac{\cos \left(2 \cdot \tan^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{z1}{z2}\right)\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{\left(z1 + z1\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= (tan (* PI (- (+ z0 z0) -0.5))) 20000000000.0)
  (/
   (-
    (* (/ 0.5 (* z1 z1)) z1)
    (/ (cos (* 2.0 (atan (* (tan (* 0.5 PI)) (/ z1 z2))))) (+ z1 z1)))
   z1)
  (/
   (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
   (* (+ z1 z1) z1))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (tan((((double) M_PI) * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((tan((0.5 * ((double) M_PI))) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (Math.tan((Math.PI * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = (((0.5 / (z1 * z1)) * z1) - (Math.cos((2.0 * Math.atan((Math.tan((0.5 * Math.PI)) * (z1 / z2))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if math.tan((math.pi * ((z0 + z0) - -0.5))) <= 20000000000.0:
		tmp = (((0.5 / (z1 * z1)) * z1) - (math.cos((2.0 * math.atan((math.tan((0.5 * math.pi)) * (z1 / z2))))) / (z1 + z1))) / z1
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / Float64(z1 * z1)) * z1) - Float64(cos(Float64(2.0 * atan(Float64(tan(Float64(0.5 * pi)) * Float64(z1 / z2))))) / Float64(z1 + z1))) / z1);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (tan((pi * ((z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = (((0.5 / (z1 * z1)) * z1) - (cos((2.0 * atan((tan((0.5 * pi)) * (z1 / z2))))) / (z1 + z1))) / z1;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 20000000000.0], N[(N[(N[(N[(0.5 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision] - N[(N[Cos[N[(2.0 * N[ArcTan[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 2e10

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 70.3%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 71.0%

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

   6. Applied rewrites 71.0%

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

   if 2e10 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 7: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan \left(\pi \cdot \left(\left(z0 + z0\right) - -0.5\right)\right) \leq 20000000000:\\ \;\;\;\;\frac{\frac{0.5}{z1} - \frac{\cos \left(-2 \cdot \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{\left(z1 + z1\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= (tan (* PI (- (+ z0 z0) -0.5))) 20000000000.0)
  (/
   (-
    (/ 0.5 z1)
    (/
     (cos (* -2.0 (atan (* (/ z1 z2) (tan (* 0.5 PI))))))
     (+ z1 z1)))
   z1)
  (/
   (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
   (* (+ z1 z1) z1))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (tan((((double) M_PI) * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = ((0.5 / z1) - (cos((-2.0 * atan(((z1 / z2) * tan((0.5 * ((double) M_PI))))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (Math.tan((Math.PI * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = ((0.5 / z1) - (Math.cos((-2.0 * Math.atan(((z1 / z2) * Math.tan((0.5 * Math.PI)))))) / (z1 + z1))) / z1;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if math.tan((math.pi * ((z0 + z0) - -0.5))) <= 20000000000.0:
		tmp = ((0.5 / z1) - (math.cos((-2.0 * math.atan(((z1 / z2) * math.tan((0.5 * math.pi)))))) / (z1 + z1))) / z1
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = Float64(Float64(Float64(0.5 / z1) - Float64(cos(Float64(-2.0 * atan(Float64(Float64(z1 / z2) * tan(Float64(0.5 * pi)))))) / Float64(z1 + z1))) / z1);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (tan((pi * ((z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = ((0.5 / z1) - (cos((-2.0 * atan(((z1 / z2) * tan((0.5 * pi)))))) / (z1 + z1))) / z1;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 20000000000.0], N[(N[(N[(0.5 / z1), $MachinePrecision] - N[(N[Cos[N[(-2.0 * N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 2e10

   1. Initial program 56.7%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 59.6%

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

   8. Applied rewrites 59.6%

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

   if 2e10 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 8: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan \left(\pi \cdot \left(\left(z0 + z0\right) - -0.5\right)\right) \leq 20000000000:\\ \;\;\;\;\frac{\frac{1 - \cos \left(\tan^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{z1}{z2}\right) \cdot 2\right)}{z1 + z1}}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{\left(z1 + z1\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (if (<= (tan (* PI (- (+ z0 z0) -0.5))) 20000000000.0)
  (/
   (/
    (- 1.0 (cos (* (atan (* (tan (* 0.5 PI)) (/ z1 z2))) 2.0)))
    (+ z1 z1))
   z1)
  (/
   (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
   (* (+ z1 z1) z1))))

C

double code(double z1, double z2, double z0) {
	double tmp;
	if (tan((((double) M_PI) * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = ((1.0 - cos((atan((tan((0.5 * ((double) M_PI))) * (z1 / z2))) * 2.0))) / (z1 + z1)) / z1;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double tmp;
	if (Math.tan((Math.PI * ((z0 + z0) - -0.5))) <= 20000000000.0) {
		tmp = ((1.0 - Math.cos((Math.atan((Math.tan((0.5 * Math.PI)) * (z1 / z2))) * 2.0))) / (z1 + z1)) / z1;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	tmp = 0
	if math.tan((math.pi * ((z0 + z0) - -0.5))) <= 20000000000.0:
		tmp = ((1.0 - math.cos((math.atan((math.tan((0.5 * math.pi)) * (z1 / z2))) * 2.0))) / (z1 + z1)) / z1
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)
	return tmp

Julia

function code(z1, z2, z0)
	tmp = 0.0
	if (tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = Float64(Float64(Float64(1.0 - cos(Float64(atan(Float64(tan(Float64(0.5 * pi)) * Float64(z1 / z2))) * 2.0))) / Float64(z1 + z1)) / z1);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	tmp = 0.0;
	if (tan((pi * ((z0 + z0) - -0.5))) <= 20000000000.0)
		tmp = ((1.0 - cos((atan((tan((0.5 * pi)) * (z1 / z2))) * 2.0))) / (z1 + z1)) / z1;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := If[LessEqual[N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 20000000000.0], N[(N[(N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(z1 + z1), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 2e10

   1. Initial program 56.7%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 59.6%

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

   if 2e10 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 9: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(z1 + z1\right) \cdot z1\\ \mathbf{if}\;\tan \left(\pi \cdot \left(\left(z0 + z0\right) - -0.5\right)\right) \leq 40:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(\frac{z1}{z2} \cdot \tan \left(\pi \cdot 0.5\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(-0.5 \cdot \frac{z1}{z0 \cdot \left(z2 \cdot \pi\right)}\right)\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (let* ((t_0 (* (+ z1 z1) z1)))
  (if (<= (tan (* PI (- (+ z0 z0) -0.5))) 40.0)
    (/
     (- 1.0 (cos (* -2.0 (atan (* (/ z1 z2) (tan (* PI 0.5)))))))
     t_0)
    (/
     (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
     t_0))))

C

double code(double z1, double z2, double z0) {
	double t_0 = (z1 + z1) * z1;
	double tmp;
	if (tan((((double) M_PI) * ((z0 + z0) - -0.5))) <= 40.0) {
		tmp = (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((((double) M_PI) * 0.5))))))) / t_0;
	} else {
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / t_0;
	}
	return tmp;
}

Java

public static double code(double z1, double z2, double z0) {
	double t_0 = (z1 + z1) * z1;
	double tmp;
	if (Math.tan((Math.PI * ((z0 + z0) - -0.5))) <= 40.0) {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan(((z1 / z2) * Math.tan((Math.PI * 0.5))))))) / t_0;
	} else {
		tmp = (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / t_0;
	}
	return tmp;
}

Python

def code(z1, z2, z0):
	t_0 = (z1 + z1) * z1
	tmp = 0
	if math.tan((math.pi * ((z0 + z0) - -0.5))) <= 40.0:
		tmp = (1.0 - math.cos((-2.0 * math.atan(((z1 / z2) * math.tan((math.pi * 0.5))))))) / t_0
	else:
		tmp = (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / t_0
	return tmp

Julia

function code(z1, z2, z0)
	t_0 = Float64(Float64(z1 + z1) * z1)
	tmp = 0.0
	if (tan(Float64(pi * Float64(Float64(z0 + z0) - -0.5))) <= 40.0)
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(Float64(z1 / z2) * tan(Float64(pi * 0.5))))))) / t_0);
	else
		tmp = Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z2, z0)
	t_0 = (z1 + z1) * z1;
	tmp = 0.0;
	if (tan((pi * ((z0 + z0) - -0.5))) <= 40.0)
		tmp = (1.0 - cos((-2.0 * atan(((z1 / z2) * tan((pi * 0.5))))))) / t_0;
	else
		tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z2_, z0_] := Block[{t$95$0 = N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]}, If[LessEqual[N[Tan[N[(Pi * N[(N[(z0 + z0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 40.0], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(N[(z1 / z2), $MachinePrecision] * N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64)))) < 40

   1. Initial program 56.7%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 57.7%

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

   if 40 < (tan.f64 (*.f64 (PI.f64) (-.f64 (+.f64 z0 z0) #s(literal -1/2 binary64))))

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 64.6%

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

   4. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 57.5%

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

   6. Applied rewrites 57.5%

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

Alternative 10: 57.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (z1 z2 z0)
  :precision binary64
  (/
 (- 1.0 (cos (* -2.0 (atan (* -0.5 (/ z1 (* z0 (* z2 PI))))))))
 (* (+ z1 z1) z1)))

C

double code(double z1, double z2, double z0) {
	return (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * ((double) M_PI))))))))) / ((z1 + z1) * z1);
}

Java

public static double code(double z1, double z2, double z0) {
	return (1.0 - Math.cos((-2.0 * Math.atan((-0.5 * (z1 / (z0 * (z2 * Math.PI)))))))) / ((z1 + z1) * z1);
}

Python

def code(z1, z2, z0):
	return (1.0 - math.cos((-2.0 * math.atan((-0.5 * (z1 / (z0 * (z2 * math.pi)))))))) / ((z1 + z1) * z1)

Julia

function code(z1, z2, z0)
	return Float64(Float64(1.0 - cos(Float64(-2.0 * atan(Float64(-0.5 * Float64(z1 / Float64(z0 * Float64(z2 * pi)))))))) / Float64(Float64(z1 + z1) * z1))
end

MATLAB

function tmp = code(z1, z2, z0)
	tmp = (1.0 - cos((-2.0 * atan((-0.5 * (z1 / (z0 * (z2 * pi)))))))) / ((z1 + z1) * z1);
end

Wolfram

code[z1_, z2_, z0_] := N[(N[(1.0 - N[Cos[N[(-2.0 * N[ArcTan[N[(-0.5 * N[(z1 / N[(z0 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 + z1), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 64.6%

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

4. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 57.5%

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

6. Applied rewrites 57.5%

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

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z2 z0)
  :name "(/ (- 1 (cos (* -2 (atan (* (/ z1 z2) (tan (* PI (- (+ z0 z0) -1/2)))))))) (* (+ z1 z1) z1))"
  :precision binary64
  (/ (- 1.0 (cos (* -2.0 (atan (* (/ z1 z2) (tan (* PI (- (+ z0 z0) -0.5)))))))) (* (+ z1 z1) z1)))