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

Percentage Accurate: 67.3% → 99.4%

Time: 6.8s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right)\\ \mathbf{if}\;z2 \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq 20000000:\\ \;\;\;\;\cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \pi \cdot 0.5\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (cos
         (*
          (atan
           (/
            (*
             (- (* (- PI (/ PI 0.0)) (+ z2 z2)) (tan (* PI -0.5)))
             z0)
            z1))
          2.0))))
  (if (<= z2 -2.8e+21)
    t_0
    (if (<= z2 20000000.0)
      (cos
       (*
        (atan
         (/
          (* (sin (+ (* (- PI) (+ z2 z2)) (* PI 0.5))) z0)
          (* (- (sin (* PI (+ z2 z2)))) z1)))
        2.0))
      t_0))))

C

double code(double z2, double z0, double z1) {
	double t_0 = cos((atan((((((((double) M_PI) - (((double) M_PI) / 0.0)) * (z2 + z2)) - tan((((double) M_PI) * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 20000000.0) {
		tmp = cos((atan(((sin(((-((double) M_PI) * (z2 + z2)) + (((double) M_PI) * 0.5))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.cos((Math.atan((((((Math.PI - (Math.PI / 0.0)) * (z2 + z2)) - Math.tan((Math.PI * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 20000000.0) {
		tmp = Math.cos((Math.atan(((Math.sin(((-Math.PI * (z2 + z2)) + (Math.PI * 0.5))) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.cos((math.atan((((((math.pi - (math.pi / 0.0)) * (z2 + z2)) - math.tan((math.pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0
	if z2 <= -2.8e+21:
		tmp = t_0
	elif z2 <= 20000000.0:
		tmp = math.cos((math.atan(((math.sin(((-math.pi * (z2 + z2)) + (math.pi * 0.5))) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1))) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = cos(Float64(atan(Float64(Float64(Float64(Float64(Float64(pi - Float64(pi / 0.0)) * Float64(z2 + z2)) - tan(Float64(pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0.0
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 20000000.0)
		tmp = cos(Float64(atan(Float64(Float64(sin(Float64(Float64(Float64(-pi) * Float64(z2 + z2)) + Float64(pi * 0.5))) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1))) * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = cos((atan((((((pi - (pi / 0.0)) * (z2 + z2)) - tan((pi * -0.5))) * z0) / z1)) * 2.0));
	tmp = 0.0;
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 20000000.0)
		tmp = cos((atan(((sin(((-pi * (z2 + z2)) + (pi * 0.5))) * z0) / (-sin((pi * (z2 + z2))) * z1))) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Cos[N[(N[ArcTan[N[(N[(N[(N[(N[(Pi - N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -2.8e+21], t$95$0, If[LessEqual[z2, 20000000.0], N[Cos[N[(N[ArcTan[N[(N[(N[Sin[N[(N[((-Pi) * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z2 < -2.8e21 or 2e7 < z2

   1. Initial program 67.3%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 78.7%

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

   6. Applied rewrites 79.2%

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 84.0%

       \[\leadsto \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \color{blue}{\left(\pi \cdot -0.5\right)}\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   if -2.8e21 < z2 < 2e7

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 76.5%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 76.5%

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

   5. Applied rewrites 76.5%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(z2 + z2\right)\\ t_1 := \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right)\\ \mathbf{if}\;z2 \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z2 \leq 20000000:\\ \;\;\;\;\cos \left(\tan^{-1} \left(\frac{\cos t\_0 \cdot z0}{\left(-\sin t\_0\right) \cdot z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* PI (+ z2 z2)))
       (t_1
        (cos
         (*
          (atan
           (/
            (*
             (- (* (- PI (/ PI 0.0)) (+ z2 z2)) (tan (* PI -0.5)))
             z0)
            z1))
          2.0))))
  (if (<= z2 -2.8e+21)
    t_1
    (if (<= z2 20000000.0)
      (cos (* (atan (/ (* (cos t_0) z0) (* (- (sin t_0)) z1))) 2.0))
      t_1))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((double) M_PI) * (z2 + z2);
	double t_1 = cos((atan((((((((double) M_PI) - (((double) M_PI) / 0.0)) * (z2 + z2)) - tan((((double) M_PI) * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_1;
	} else if (z2 <= 20000000.0) {
		tmp = cos((atan(((cos(t_0) * z0) / (-sin(t_0) * z1))) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.PI * (z2 + z2);
	double t_1 = Math.cos((Math.atan((((((Math.PI - (Math.PI / 0.0)) * (z2 + z2)) - Math.tan((Math.PI * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_1;
	} else if (z2 <= 20000000.0) {
		tmp = Math.cos((Math.atan(((Math.cos(t_0) * z0) / (-Math.sin(t_0) * z1))) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pi * (z2 + z2)
	t_1 = math.cos((math.atan((((((math.pi - (math.pi / 0.0)) * (z2 + z2)) - math.tan((math.pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0
	if z2 <= -2.8e+21:
		tmp = t_1
	elif z2 <= 20000000.0:
		tmp = math.cos((math.atan(((math.cos(t_0) * z0) / (-math.sin(t_0) * z1))) * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(pi * Float64(z2 + z2))
	t_1 = cos(Float64(atan(Float64(Float64(Float64(Float64(Float64(pi - Float64(pi / 0.0)) * Float64(z2 + z2)) - tan(Float64(pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0.0
	if (z2 <= -2.8e+21)
		tmp = t_1;
	elseif (z2 <= 20000000.0)
		tmp = cos(Float64(atan(Float64(Float64(cos(t_0) * z0) / Float64(Float64(-sin(t_0)) * z1))) * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = pi * (z2 + z2);
	t_1 = cos((atan((((((pi - (pi / 0.0)) * (z2 + z2)) - tan((pi * -0.5))) * z0) / z1)) * 2.0));
	tmp = 0.0;
	if (z2 <= -2.8e+21)
		tmp = t_1;
	elseif (z2 <= 20000000.0)
		tmp = cos((atan(((cos(t_0) * z0) / (-sin(t_0) * z1))) * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[ArcTan[N[(N[(N[(N[(N[(Pi - N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -2.8e+21], t$95$1, If[LessEqual[z2, 20000000.0], N[Cos[N[(N[ArcTan[N[(N[(N[Cos[t$95$0], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[t$95$0], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if z2 < -2.8e21 or 2e7 < z2

   1. Initial program 67.3%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 78.7%

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

   6. Applied rewrites 79.2%

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 84.0%

       \[\leadsto \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \color{blue}{\left(\pi \cdot -0.5\right)}\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   if -2.8e21 < z2 < 2e7

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 76.5%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right)\\ \mathbf{if}\;z2 \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq 20000000:\\ \;\;\;\;\cos \left(\tan^{-1} \left(\frac{z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (cos
         (*
          (atan
           (/
            (*
             (- (* (- PI (/ PI 0.0)) (+ z2 z2)) (tan (* PI -0.5)))
             z0)
            z1))
          2.0))))
  (if (<= z2 -2.8e+21)
    t_0
    (if (<= z2 20000000.0)
      (cos (* (atan (/ z0 (* (- (sin (* PI (+ z2 z2)))) z1))) 2.0))
      t_0))))

C

double code(double z2, double z0, double z1) {
	double t_0 = cos((atan((((((((double) M_PI) - (((double) M_PI) / 0.0)) * (z2 + z2)) - tan((((double) M_PI) * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 20000000.0) {
		tmp = cos((atan((z0 / (-sin((((double) M_PI) * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.cos((Math.atan((((((Math.PI - (Math.PI / 0.0)) * (z2 + z2)) - Math.tan((Math.PI * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 20000000.0) {
		tmp = Math.cos((Math.atan((z0 / (-Math.sin((Math.PI * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.cos((math.atan((((((math.pi - (math.pi / 0.0)) * (z2 + z2)) - math.tan((math.pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0
	if z2 <= -2.8e+21:
		tmp = t_0
	elif z2 <= 20000000.0:
		tmp = math.cos((math.atan((z0 / (-math.sin((math.pi * (z2 + z2))) * z1))) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = cos(Float64(atan(Float64(Float64(Float64(Float64(Float64(pi - Float64(pi / 0.0)) * Float64(z2 + z2)) - tan(Float64(pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0.0
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 20000000.0)
		tmp = cos(Float64(atan(Float64(z0 / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1))) * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = cos((atan((((((pi - (pi / 0.0)) * (z2 + z2)) - tan((pi * -0.5))) * z0) / z1)) * 2.0));
	tmp = 0.0;
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 20000000.0)
		tmp = cos((atan((z0 / (-sin((pi * (z2 + z2))) * z1))) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Cos[N[(N[ArcTan[N[(N[(N[(N[(N[(Pi - N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -2.8e+21], t$95$0, If[LessEqual[z2, 20000000.0], N[Cos[N[(N[ArcTan[N[(z0 / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z2 < -2.8e21 or 2e7 < z2

   1. Initial program 67.3%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 78.7%

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

   6. Applied rewrites 79.2%

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 84.0%

       \[\leadsto \cos \left(\tan^{-1} \left(\frac{\left(\left(\pi - \frac{\pi}{0}\right) \cdot \left(z2 + z2\right) - \tan \color{blue}{\left(\pi \cdot -0.5\right)}\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   if -2.8e21 < z2 < 2e7

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 76.5%

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

Alternative 4: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \leq 100:\\ \;\;\;\;\cos \left(\tan^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\tan^{-1} \left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= (tan (* (- (+ z2 z2) -0.5) PI)) 100.0)
  (cos (* (atan (* (tan (* 0.5 PI)) (/ z0 z1))) 2.0))
  (cos (* (atan (* -0.5 (/ z0 (* z1 (* z2 PI))))) 2.0))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (tan((((z2 + z2) - -0.5) * ((double) M_PI))) <= 100.0) {
		tmp = cos((atan((tan((0.5 * ((double) M_PI))) * (z0 / z1))) * 2.0));
	} else {
		tmp = cos((atan((-0.5 * (z0 / (z1 * (z2 * ((double) M_PI)))))) * 2.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (Math.tan((((z2 + z2) - -0.5) * Math.PI)) <= 100.0) {
		tmp = Math.cos((Math.atan((Math.tan((0.5 * Math.PI)) * (z0 / z1))) * 2.0));
	} else {
		tmp = Math.cos((Math.atan((-0.5 * (z0 / (z1 * (z2 * Math.PI))))) * 2.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if math.tan((((z2 + z2) - -0.5) * math.pi)) <= 100.0:
		tmp = math.cos((math.atan((math.tan((0.5 * math.pi)) * (z0 / z1))) * 2.0))
	else:
		tmp = math.cos((math.atan((-0.5 * (z0 / (z1 * (z2 * math.pi))))) * 2.0))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) <= 100.0)
		tmp = cos(Float64(atan(Float64(tan(Float64(0.5 * pi)) * Float64(z0 / z1))) * 2.0));
	else
		tmp = cos(Float64(atan(Float64(-0.5 * Float64(z0 / Float64(z1 * Float64(z2 * pi))))) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (tan((((z2 + z2) - -0.5) * pi)) <= 100.0)
		tmp = cos((atan((tan((0.5 * pi)) * (z0 / z1))) * 2.0));
	else
		tmp = cos((atan((-0.5 * (z0 / (z1 * (z2 * pi))))) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 100.0], N[Cos[N[(N[ArcTan[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Cos[N[(N[ArcTan[N[(-0.5 * N[(z0 / N[(z1 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) < 100

   1. Initial program 67.3%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 68.6%

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

   if 100 < (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)))

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 76.5%

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

   4. Taylor expanded in z2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 66.2%

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

   6. Applied rewrites 66.2%

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

Alternative 5: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 76.5%

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

4. Taylor expanded in z2 around 0

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

6. Applied rewrites 76.5%

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

Alternative 6: 66.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 76.5%

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

4. Taylor expanded in z2 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 66.2%

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

6. Applied rewrites 66.2%

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

Reproduce

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