(/ (sin (* z1 (* PI z0))) (* z1 (* PI z0)))

Percentage Accurate: 42.3% → 97.0%

Time: 2.8s

Alternatives: 5

Speedup: 132.0×

Specification

Math

\[\begin{array}{l} t_0 := z1 \cdot \left(\pi \cdot z0\right)\\ \frac{\sin t\_0}{t\_0} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* z1 (* PI z0)))) (/ (sin t_0) t_0)))

C

double code(double z1, double z0) {
	double t_0 = z1 * (((double) M_PI) * z0);
	return sin(t_0) / t_0;
}

Java

public static double code(double z1, double z0) {
	double t_0 = z1 * (Math.PI * z0);
	return Math.sin(t_0) / t_0;
}

Python

def code(z1, z0):
	t_0 = z1 * (math.pi * z0)
	return math.sin(t_0) / t_0

Julia

function code(z1, z0)
	t_0 = Float64(z1 * Float64(pi * z0))
	return Float64(sin(t_0) / t_0)
end

MATLAB

function tmp = code(z1, z0)
	t_0 = z1 * (pi * z0);
	tmp = sin(t_0) / t_0;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(z1 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision]}, N[(N[Sin[t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]

Initial Program: 42.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := z1 \cdot \left(\pi \cdot z0\right)\\ \frac{\sin t\_0}{t\_0} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* z1 (* PI z0)))) (/ (sin t_0) t_0)))

C

double code(double z1, double z0) {
	double t_0 = z1 * (((double) M_PI) * z0);
	return sin(t_0) / t_0;
}

Java

public static double code(double z1, double z0) {
	double t_0 = z1 * (Math.PI * z0);
	return Math.sin(t_0) / t_0;
}

Python

def code(z1, z0):
	t_0 = z1 * (math.pi * z0)
	return math.sin(t_0) / t_0

Julia

function code(z1, z0)
	t_0 = Float64(z1 * Float64(pi * z0))
	return Float64(sin(t_0) / t_0)
end

MATLAB

function tmp = code(z1, z0)
	t_0 = z1 * (pi * z0);
	tmp = sin(t_0) / t_0;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(z1 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision]}, N[(N[Sin[t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]

Alternative 1: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left|z0\right| \cdot \left|z1\right|\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\\ t_1 := \left|z1\right| \cdot \left(\pi \cdot \left|z0\right|\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\sin t\_0}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\pi \cdot \left|z1\right|\right) \cdot \left|z0\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0
        (*
         (* (* (fabs z0) (fabs z1)) (pow PI 0.6666666666666666))
         (cbrt PI)))
       (t_1 (* (fabs z1) (* PI (fabs z0)))))
  (if (<= t_1 5e-324)
    1.0
    (if (<= t_1 5e+28)
      (/ (sin t_0) t_0)
      (/ 0.0 (* (* PI (fabs z1)) (fabs z0)))))))

C

double code(double z1, double z0) {
	double t_0 = ((fabs(z0) * fabs(z1)) * pow(((double) M_PI), 0.6666666666666666)) * cbrt(((double) M_PI));
	double t_1 = fabs(z1) * (((double) M_PI) * fabs(z0));
	double tmp;
	if (t_1 <= 5e-324) {
		tmp = 1.0;
	} else if (t_1 <= 5e+28) {
		tmp = sin(t_0) / t_0;
	} else {
		tmp = 0.0 / ((((double) M_PI) * fabs(z1)) * fabs(z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = ((Math.abs(z0) * Math.abs(z1)) * Math.pow(Math.PI, 0.6666666666666666)) * Math.cbrt(Math.PI);
	double t_1 = Math.abs(z1) * (Math.PI * Math.abs(z0));
	double tmp;
	if (t_1 <= 5e-324) {
		tmp = 1.0;
	} else if (t_1 <= 5e+28) {
		tmp = Math.sin(t_0) / t_0;
	} else {
		tmp = 0.0 / ((Math.PI * Math.abs(z1)) * Math.abs(z0));
	}
	return tmp;
}

Julia

function code(z1, z0)
	t_0 = Float64(Float64(Float64(abs(z0) * abs(z1)) * (pi ^ 0.6666666666666666)) * cbrt(pi))
	t_1 = Float64(abs(z1) * Float64(pi * abs(z0)))
	tmp = 0.0
	if (t_1 <= 5e-324)
		tmp = 1.0;
	elseif (t_1 <= 5e+28)
		tmp = Float64(sin(t_0) / t_0);
	else
		tmp = Float64(0.0 / Float64(Float64(pi * abs(z1)) * abs(z0)));
	end
	return tmp
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z1], $MachinePrecision] * N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-324], 1.0, If[LessEqual[t$95$1, 5e+28], N[(N[Sin[t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.0 / N[(N[(Pi * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z1 (*.f64 (PI.f64) z0)) < 4.9406564584124654e-324

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{1} \]

   if 4.9406564584124654e-324 < (*.f64 z1 (*.f64 (PI.f64) z0)) < 4.9999999999999996e28

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(z1 \cdot \left(\pi \cdot z0\right)\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \color{blue}{\left(\pi \cdot z0\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \color{blue}{\left(z0 \cdot \pi\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(z1 \cdot z0\right) \cdot \pi\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto \frac{\sin \left(\left(z1 \cdot z0\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\sin \left(\left(z1 \cdot z0\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      13. pow1/3 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot \left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      14. lift-PI.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot \left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      15. pow1/3 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot \left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      16. pow-prod-up N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      19. lift-PI.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      20. lower-cbrt.f64 41.5%

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   3. Applied rewrites 41.5%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{z1 \cdot \left(\pi \cdot z0\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{z1 \cdot \color{blue}{\left(\pi \cdot z0\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{z1 \cdot \color{blue}{\left(z0 \cdot \pi\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{\left(z1 \cdot z0\right) \cdot \pi}} \]

      5. lift-PI.f64 N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(z1 \cdot z0\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

      6. add-cube-cbrt N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(z1 \cdot z0\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \]

      12. lift-PI.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\left(z0 \cdot z1\right) \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \]

      13. pow1/3 N/A

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

      14. lift-PI.f64 N/A

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

      15. pow1/3 N/A

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

      16. pow-prod-up N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \]

      19. lift-PI.f64 N/A

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\left(z0 \cdot z1\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\color{blue}{\pi}}} \]

      20. lower-cbrt.f64 42.2%

          \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)}{\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \color{blue}{\sqrt[3]{\pi}}} \]

   5. Applied rewrites 42.2%

      \[\leadsto \frac{\sin \left(\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)}{\color{blue}{\left(\left(z0 \cdot z1\right) \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}}} \]

   if 4.9999999999999996e28 < (*.f64 z1 (*.f64 (PI.f64) z0))

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]
   2. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot 1}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)} \cdot 1}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 6.2%

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

      1. lift-neg.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sin-neg-rev N/A

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

      14. cos-+PI/2-rev N/A

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 47.2%

         \[\leadsto \frac{0.5 \cdot \left(\cos \left(-0.5 \cdot \pi\right) - \cos \left(0.5 \cdot \pi\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   8. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      8. metadata-eval 47.2%

         \[\leadsto \frac{0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{0}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z1 \cdot \left(\pi \cdot z0\right)\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{0}{z1 \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\pi \cdot z0\right)\right)} \]

   10. Applied rewrites 47.2%

       \[\leadsto \color{blue}{\frac{0}{\left(\pi \cdot z1\right) \cdot z0}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z1\right|, \left|z0\right|\right)\\ t_1 := \mathsf{max}\left(\left|z1\right|, \left|z0\right|\right)\\ t_2 := t\_0 \cdot \left(\pi \cdot t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\sin t\_2}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\pi \cdot t\_0\right) \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (fmin (fabs z1) (fabs z0)))
       (t_1 (fmax (fabs z1) (fabs z0)))
       (t_2 (* t_0 (* PI t_1))))
  (if (<= t_2 0.0)
    1.0
    (if (<= t_2 5e+28) (/ (sin t_2) t_2) (/ 0.0 (* (* PI t_0) t_1))))))

C

double code(double z1, double z0) {
	double t_0 = fmin(fabs(z1), fabs(z0));
	double t_1 = fmax(fabs(z1), fabs(z0));
	double t_2 = t_0 * (((double) M_PI) * t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+28) {
		tmp = sin(t_2) / t_2;
	} else {
		tmp = 0.0 / ((((double) M_PI) * t_0) * t_1);
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = fmin(Math.abs(z1), Math.abs(z0));
	double t_1 = fmax(Math.abs(z1), Math.abs(z0));
	double t_2 = t_0 * (Math.PI * t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+28) {
		tmp = Math.sin(t_2) / t_2;
	} else {
		tmp = 0.0 / ((Math.PI * t_0) * t_1);
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = fmin(math.fabs(z1), math.fabs(z0))
	t_1 = fmax(math.fabs(z1), math.fabs(z0))
	t_2 = t_0 * (math.pi * t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = 1.0
	elif t_2 <= 5e+28:
		tmp = math.sin(t_2) / t_2
	else:
		tmp = 0.0 / ((math.pi * t_0) * t_1)
	return tmp

Julia

function code(z1, z0)
	t_0 = fmin(abs(z1), abs(z0))
	t_1 = fmax(abs(z1), abs(z0))
	t_2 = Float64(t_0 * Float64(pi * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+28)
		tmp = Float64(sin(t_2) / t_2);
	else
		tmp = Float64(0.0 / Float64(Float64(pi * t_0) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = min(abs(z1), abs(z0));
	t_1 = max(abs(z1), abs(z0));
	t_2 = t_0 * (pi * t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+28)
		tmp = sin(t_2) / t_2;
	else
		tmp = 0.0 / ((pi * t_0) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[Min[N[Abs[z1], $MachinePrecision], N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z1], $MachinePrecision], N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], 1.0, If[LessEqual[t$95$2, 5e+28], N[(N[Sin[t$95$2], $MachinePrecision] / t$95$2), $MachinePrecision], N[(0.0 / N[(N[(Pi * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z1 (*.f64 (PI.f64) z0)) < 0.0

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (*.f64 z1 (*.f64 (PI.f64) z0)) < 4.9999999999999996e28

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   if 4.9999999999999996e28 < (*.f64 z1 (*.f64 (PI.f64) z0))

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]
   2. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot 1}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)} \cdot 1}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 6.2%

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

      1. lift-neg.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sin-neg-rev N/A

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

      14. cos-+PI/2-rev N/A

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 47.2%

         \[\leadsto \frac{0.5 \cdot \left(\cos \left(-0.5 \cdot \pi\right) - \cos \left(0.5 \cdot \pi\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   8. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      8. metadata-eval 47.2%

         \[\leadsto \frac{0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{0}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z1 \cdot \left(\pi \cdot z0\right)\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{0}{z1 \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\pi \cdot z0\right)\right)} \]

   10. Applied rewrites 47.2%

       \[\leadsto \color{blue}{\frac{0}{\left(\pi \cdot z1\right) \cdot z0}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|z1\right| \cdot \left(\pi \cdot \left|z0\right|\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\sin \left(\left|z0\right| \cdot \left(\left|z1\right| \cdot \pi\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\pi \cdot \left|z1\right|\right) \cdot \left|z0\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (fabs z1) (* PI (fabs z0)))))
  (if (<= t_0 5e-12)
    1.0
    (if (<= t_0 5e+28)
      (/ (sin (* (fabs z0) (* (fabs z1) PI))) t_0)
      (/ 0.0 (* (* PI (fabs z1)) (fabs z0)))))))

C

double code(double z1, double z0) {
	double t_0 = fabs(z1) * (((double) M_PI) * fabs(z0));
	double tmp;
	if (t_0 <= 5e-12) {
		tmp = 1.0;
	} else if (t_0 <= 5e+28) {
		tmp = sin((fabs(z0) * (fabs(z1) * ((double) M_PI)))) / t_0;
	} else {
		tmp = 0.0 / ((((double) M_PI) * fabs(z1)) * fabs(z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.abs(z1) * (Math.PI * Math.abs(z0));
	double tmp;
	if (t_0 <= 5e-12) {
		tmp = 1.0;
	} else if (t_0 <= 5e+28) {
		tmp = Math.sin((Math.abs(z0) * (Math.abs(z1) * Math.PI))) / t_0;
	} else {
		tmp = 0.0 / ((Math.PI * Math.abs(z1)) * Math.abs(z0));
	}
	return tmp;
}

Python

def code(z1, z0):
	t_0 = math.fabs(z1) * (math.pi * math.fabs(z0))
	tmp = 0
	if t_0 <= 5e-12:
		tmp = 1.0
	elif t_0 <= 5e+28:
		tmp = math.sin((math.fabs(z0) * (math.fabs(z1) * math.pi))) / t_0
	else:
		tmp = 0.0 / ((math.pi * math.fabs(z1)) * math.fabs(z0))
	return tmp

Julia

function code(z1, z0)
	t_0 = Float64(abs(z1) * Float64(pi * abs(z0)))
	tmp = 0.0
	if (t_0 <= 5e-12)
		tmp = 1.0;
	elseif (t_0 <= 5e+28)
		tmp = Float64(sin(Float64(abs(z0) * Float64(abs(z1) * pi))) / t_0);
	else
		tmp = Float64(0.0 / Float64(Float64(pi * abs(z1)) * abs(z0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = abs(z1) * (pi * abs(z0));
	tmp = 0.0;
	if (t_0 <= 5e-12)
		tmp = 1.0;
	elseif (t_0 <= 5e+28)
		tmp = sin((abs(z0) * (abs(z1) * pi))) / t_0;
	else
		tmp = 0.0 / ((pi * abs(z1)) * abs(z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[Abs[z1], $MachinePrecision] * N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-12], 1.0, If[LessEqual[t$95$0, 5e+28], N[(N[Sin[N[(N[Abs[z0], $MachinePrecision] * N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.0 / N[(N[(Pi * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z1 (*.f64 (PI.f64) z0)) < 4.9999999999999997e-12

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{1} \]

   if 4.9999999999999997e-12 < (*.f64 z1 (*.f64 (PI.f64) z0)) < 4.9999999999999996e28

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   2. Taylor expanded in z1 around inf

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

      1. lower-sin.f64 N/A

         \[\leadsto \frac{\sin \left(z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(z0 \cdot \left(z1 \cdot \mathsf{PI}\left(\right)\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. lower-PI.f64 42.1%

         \[\leadsto \frac{\sin \left(z0 \cdot \left(z1 \cdot \pi\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   4. Applied rewrites 42.1%

      \[\leadsto \frac{\color{blue}{\sin \left(z0 \cdot \left(z1 \cdot \pi\right)\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

   if 4.9999999999999996e28 < (*.f64 z1 (*.f64 (PI.f64) z0))

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]
   2. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot 1}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)} \cdot 1}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 6.2%

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

      1. lift-neg.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sin-neg-rev N/A

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

      14. cos-+PI/2-rev N/A

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 47.2%

         \[\leadsto \frac{0.5 \cdot \left(\cos \left(-0.5 \cdot \pi\right) - \cos \left(0.5 \cdot \pi\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   8. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      8. metadata-eval 47.2%

         \[\leadsto \frac{0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{0}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z1 \cdot \left(\pi \cdot z0\right)\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{0}{z1 \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\pi \cdot z0\right)\right)} \]

   10. Applied rewrites 47.2%

       \[\leadsto \color{blue}{\frac{0}{\left(\pi \cdot z1\right) \cdot z0}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z1\right| \cdot \left(\pi \cdot \left|z0\right|\right) \leq 2 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\left(\pi \cdot \left|z1\right|\right) \cdot \left|z0\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= (* (fabs z1) (* PI (fabs z0))) 2e+27)
  1.0
  (/ 0.0 (* (* PI (fabs z1)) (fabs z0)))))

C

double code(double z1, double z0) {
	double tmp;
	if ((fabs(z1) * (((double) M_PI) * fabs(z0))) <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = 0.0 / ((((double) M_PI) * fabs(z1)) * fabs(z0));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if ((Math.abs(z1) * (Math.PI * Math.abs(z0))) <= 2e+27) {
		tmp = 1.0;
	} else {
		tmp = 0.0 / ((Math.PI * Math.abs(z1)) * Math.abs(z0));
	}
	return tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if (math.fabs(z1) * (math.pi * math.fabs(z0))) <= 2e+27:
		tmp = 1.0
	else:
		tmp = 0.0 / ((math.pi * math.fabs(z1)) * math.fabs(z0))
	return tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(abs(z1) * Float64(pi * abs(z0))) <= 2e+27)
		tmp = 1.0;
	else
		tmp = Float64(0.0 / Float64(Float64(pi * abs(z1)) * abs(z0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if ((abs(z1) * (pi * abs(z0))) <= 2e+27)
		tmp = 1.0;
	else
		tmp = 0.0 / ((pi * abs(z1)) * abs(z0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[(N[Abs[z1], $MachinePrecision] * N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+27], 1.0, N[(0.0 / N[(N[(Pi * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z1 (*.f64 (PI.f64) z0)) < 2e27

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{1} \]

   if 2e27 < (*.f64 z1 (*.f64 (PI.f64) z0))

   1. Initial program 42.3%

      \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]
   2. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot 1}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)} \cdot 1}{z1 \cdot \left(\pi \cdot z0\right)} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{z1 \cdot \left(\pi \cdot z0\right)} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 6.2%

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

      1. lift-neg.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. sin-neg-rev N/A

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

      14. cos-+PI/2-rev N/A

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 47.2%

         \[\leadsto \frac{0.5 \cdot \left(\cos \left(-0.5 \cdot \pi\right) - \cos \left(0.5 \cdot \pi\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

   8. Applied rewrites 47.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lift-cos.f64 N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{\frac{1}{2} \cdot 0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      8. metadata-eval 47.2%

         \[\leadsto \frac{0}{z1 \cdot \left(\pi \cdot z0\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{0}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(z1 \cdot \left(\pi \cdot z0\right)\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{0}{z1 \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\pi \cdot z0\right)\right)} \]

   10. Applied rewrites 47.2%

       \[\leadsto \color{blue}{\frac{0}{\left(\pi \cdot z1\right) \cdot z0}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.9% accurate, 132.0× speedup

Math

\[1 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  1.0)

C

double code(double z1, double z0) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 1.0d0
end function

Java

public static double code(double z1, double z0) {
	return 1.0;
}

Python

def code(z1, z0):
	return 1.0

Julia

function code(z1, z0)
	return 1.0
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0;
end

Wolfram

code[z1_, z0_] := 1.0

Derivation

1. Initial program 42.3%

   \[\frac{\sin \left(z1 \cdot \left(\pi \cdot z0\right)\right)}{z1 \cdot \left(\pi \cdot z0\right)} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(/ (sin (* z1 (* PI z0))) (* z1 (* PI z0)))"
  :precision binary64
  (/ (sin (* z1 (* PI z0))) (* z1 (* PI z0))))