(sin (* (+ PI PI) z0))

Percentage Accurate: 52.9% → 98.5%

Time: 18.1s

Alternatives: 6

Speedup: 0.3×

Specification

Math

\[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (+ PI PI) z0)))

C

double code(double z0) {
	return sin(((((double) M_PI) + ((double) M_PI)) * z0));
}

Java

public static double code(double z0) {
	return Math.sin(((Math.PI + Math.PI) * z0));
}

Python

def code(z0):
	return math.sin(((math.pi + math.pi) * z0))

Julia

function code(z0)
	return sin(Float64(Float64(pi + pi) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((pi + pi) * z0));
end

Wolfram

code[z0_] := N[Sin[N[(N[(Pi + Pi), $MachinePrecision] * z0), $MachinePrecision]], $MachinePrecision]

Initial Program: 52.9% accurate, 1.0× speedup

Math

\[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (+ PI PI) z0)))

C

double code(double z0) {
	return sin(((((double) M_PI) + ((double) M_PI)) * z0));
}

Java

public static double code(double z0) {
	return Math.sin(((Math.PI + Math.PI) * z0));
}

Python

def code(z0):
	return math.sin(((math.pi + math.pi) * z0))

Julia

function code(z0)
	return sin(Float64(Float64(pi + pi) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((pi + pi) * z0));
end

Wolfram

code[z0_] := N[Sin[N[(N[(Pi + Pi), $MachinePrecision] * z0), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\frac{1}{2} + \left|z0\right|\right)\\ t_1 := t\_0 + t\_0\\ t_2 := \pi \cdot \left(\left|z0\right| - \frac{1}{2}\right)\\ t_3 := t\_2 - t\_2\\ t_4 := \left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq \frac{3602879701896397}{72057594037927936}:\\ \;\;\;\;\sin \left(\left(\left(2 \cdot \left|z0\right|\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 2, \left(\sin \left(\frac{t\_3 + t\_4}{2}\right) \cdot \cos \left(\frac{t\_3 - t\_4}{2}\right)\right), 2, \left(\sin \left(\frac{t\_1 - t\_3}{2}\right) \cdot \cos \left(\frac{t\_1 + t\_3}{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (+ 1/2 (fabs z0))))
       (t_1 (+ t_0 t_0))
       (t_2 (* PI (- (fabs z0) 1/2)))
       (t_3 (- t_2 t_2))
       (t_4 (* (+ (fabs z0) (fabs z0)) PI)))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 3602879701896397/72057594037927936)
     (sin (* (* (* 2 (fabs z0)) (pow PI 2/3)) (pow (pow PI 1/6) 2)))
     (304-z0z1z2z3z4
      1/2
      2
      (* (sin (/ (+ t_3 t_4) 2)) (cos (/ (- t_3 t_4) 2)))
      2
      (* (sin (/ (- t_1 t_3) 2)) (cos (/ (+ t_1 t_3) 2))))))))

Derivation

1. Split input into 2 regimes

2. if z0 < 0.050000000000000003

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*r* N/A

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

      6. lift-PI.f64 N/A

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

      7. add-cube-cbrt N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

      7. lower-unsound-pow.f64 N/A

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

         \[\leadsto \sin \left(\left(\left(2 \cdot z0\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\color{blue}{\frac{1}{6}}}\right)}^{2}\right) \]

   7. Applied rewrites 52.8%

      \[\leadsto \sin \left(\left(\left(2 \cdot z0\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \color{blue}{{\left({\pi}^{\frac{1}{6}}\right)}^{2}}\right) \]

   if 0.050000000000000003 < z0

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

      14. lower-*.f64 52.9%

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   3. Applied rewrites 52.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 2, \left(\sin \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) + \left(z0 + z0\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) - \left(z0 + z0\right) \cdot \pi}{2}\right)\right), 2, \left(\sin \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) - \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) + \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq \frac{3602879701896397}{72057594037927936}:\\ \;\;\;\;\sin \left(\left(\left(2 \cdot \left|z0\right|\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\sin \left(\left(\pi + \pi\right) \cdot \left(\left|z0\right| - \frac{-1}{2}\right)\right) - \sin \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1 z0)
 (if (<= (fabs z0) 3602879701896397/72057594037927936)
   (sin (* (* (* 2 (fabs z0)) (pow PI 2/3)) (pow (pow PI 1/6) 2)))
   (*
    (-
     (-
      (sin (* (+ PI PI) (- (fabs z0) -1/2)))
      (sin (* (+ (fabs z0) (fabs z0)) PI))))
    1/2))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 0.05) {
		tmp = sin((((2.0 * fabs(z0)) * pow(((double) M_PI), 0.6666666666666666)) * pow(pow(((double) M_PI), 0.16666666666666666), 2.0)));
	} else {
		tmp = -(sin(((((double) M_PI) + ((double) M_PI)) * (fabs(z0) - -0.5))) - sin(((fabs(z0) + fabs(z0)) * ((double) M_PI)))) * 0.5;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 0.05) {
		tmp = Math.sin((((2.0 * Math.abs(z0)) * Math.pow(Math.PI, 0.6666666666666666)) * Math.pow(Math.pow(Math.PI, 0.16666666666666666), 2.0)));
	} else {
		tmp = -(Math.sin(((Math.PI + Math.PI) * (Math.abs(z0) - -0.5))) - Math.sin(((Math.abs(z0) + Math.abs(z0)) * Math.PI))) * 0.5;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 0.05:
		tmp = math.sin((((2.0 * math.fabs(z0)) * math.pow(math.pi, 0.6666666666666666)) * math.pow(math.pow(math.pi, 0.16666666666666666), 2.0)))
	else:
		tmp = -(math.sin(((math.pi + math.pi) * (math.fabs(z0) - -0.5))) - math.sin(((math.fabs(z0) + math.fabs(z0)) * math.pi))) * 0.5
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 0.05)
		tmp = sin(Float64(Float64(Float64(2.0 * abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	else
		tmp = Float64(Float64(-Float64(sin(Float64(Float64(pi + pi) * Float64(abs(z0) - -0.5))) - sin(Float64(Float64(abs(z0) + abs(z0)) * pi)))) * 0.5);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 0.05)
		tmp = sin((((2.0 * abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	else
		tmp = -(sin(((pi + pi) * (abs(z0) - -0.5))) - sin(((abs(z0) + abs(z0)) * pi))) * 0.5;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 3602879701896397/72057594037927936], N[Sin[N[(N[(N[(2 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 2/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[Pi, 1/6], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[((-N[(N[Sin[N[(N[(Pi + Pi), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] - -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Sin[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) * 1/2), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 0.050000000000000003

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*r* N/A

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

      6. lift-PI.f64 N/A

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

      7. add-cube-cbrt N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

      7. lower-unsound-pow.f64 N/A

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

         \[\leadsto \sin \left(\left(\left(2 \cdot z0\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\color{blue}{\frac{1}{6}}}\right)}^{2}\right) \]

   7. Applied rewrites 52.8%

      \[\leadsto \sin \left(\left(\left(2 \cdot z0\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \color{blue}{{\left({\pi}^{\frac{1}{6}}\right)}^{2}}\right) \]

   if 0.050000000000000003 < z0

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

      14. lower-*.f64 52.9%

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   3. Applied rewrites 52.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 2, \left(\sin \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) + \left(z0 + z0\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) - \left(z0 + z0\right) \cdot \pi}{2}\right)\right), 2, \left(\sin \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) - \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) + \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right)\right)\right)} \]

   9. Applied rewrites 52.1%

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

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 13500000:\\ \;\;\;\;2 \cdot \left(\sin t\_0 \cdot \sin \left(\pi \cdot \frac{1}{2} + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\sin \left(\left(\pi + \pi\right) \cdot \left(\left|z0\right| - \frac{-1}{2}\right)\right) - \sin \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI)))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 13500000)
     (* 2 (* (sin t_0) (sin (+ (* PI 1/2) t_0))))
     (*
      (-
       (-
        (sin (* (+ PI PI) (- (fabs z0) -1/2)))
        (sin (* (+ (fabs z0) (fabs z0)) PI))))
      1/2)))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 13500000.0) {
		tmp = 2.0 * (sin(t_0) * sin(((((double) M_PI) * 0.5) + t_0)));
	} else {
		tmp = -(sin(((((double) M_PI) + ((double) M_PI)) * (fabs(z0) - -0.5))) - sin(((fabs(z0) + fabs(z0)) * ((double) M_PI)))) * 0.5;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 13500000.0) {
		tmp = 2.0 * (Math.sin(t_0) * Math.sin(((Math.PI * 0.5) + t_0)));
	} else {
		tmp = -(Math.sin(((Math.PI + Math.PI) * (Math.abs(z0) - -0.5))) - Math.sin(((Math.abs(z0) + Math.abs(z0)) * Math.PI))) * 0.5;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	tmp = 0
	if math.fabs(z0) <= 13500000.0:
		tmp = 2.0 * (math.sin(t_0) * math.sin(((math.pi * 0.5) + t_0)))
	else:
		tmp = -(math.sin(((math.pi + math.pi) * (math.fabs(z0) - -0.5))) - math.sin(((math.fabs(z0) + math.fabs(z0)) * math.pi))) * 0.5
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	tmp = 0.0
	if (abs(z0) <= 13500000.0)
		tmp = Float64(2.0 * Float64(sin(t_0) * sin(Float64(Float64(pi * 0.5) + t_0))));
	else
		tmp = Float64(Float64(-Float64(sin(Float64(Float64(pi + pi) * Float64(abs(z0) - -0.5))) - sin(Float64(Float64(abs(z0) + abs(z0)) * pi)))) * 0.5);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) * pi;
	tmp = 0.0;
	if (abs(z0) <= 13500000.0)
		tmp = 2.0 * (sin(t_0) * sin(((pi * 0.5) + t_0)));
	else
		tmp = -(sin(((pi + pi) * (abs(z0) - -0.5))) - sin(((abs(z0) + abs(z0)) * pi))) * 0.5;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 13500000], N[(2 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(N[(Pi * 1/2), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(N[Sin[N[(N[(Pi + Pi), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] - -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Sin[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) * 1/2), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.35e7

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

      14. lower-*.f64 52.9%

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   3. Applied rewrites 52.9%

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

      1. lift-cos.f64 N/A

         \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 52.8%

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

   5. Applied rewrites 52.8%

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

   if 1.35e7 < z0

   1. Initial program 52.9%

      \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

      14. lower-*.f64 52.9%

          \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   3. Applied rewrites 52.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 2, \left(\sin \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) + \left(z0 + z0\right) \cdot \pi}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right) - \left(z0 + z0\right) \cdot \pi}{2}\right)\right), 2, \left(\sin \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) - \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(\pi \cdot \left(\frac{1}{2} + z0\right) + \pi \cdot \left(\frac{1}{2} + z0\right)\right) + \left(\pi \cdot \left(z0 - \frac{1}{2}\right) - \pi \cdot \left(z0 - \frac{1}{2}\right)\right)}{2}\right)\right)\right)} \]

   9. Applied rewrites 52.1%

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

Alternative 4: 52.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \left(2 \cdot \left(\sin t\_0 \cdot \sin \left(\pi \cdot \frac{1}{2} + t\_0\right)\right)\right) \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI)))
  (* (copysign 1 z0) (* 2 (* (sin t_0) (sin (+ (* PI 1/2) t_0)))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	return copysign(1.0, z0) * (2.0 * (sin(t_0) * sin(((((double) M_PI) * 0.5) + t_0))));
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	return Math.copySign(1.0, z0) * (2.0 * (Math.sin(t_0) * Math.sin(((Math.PI * 0.5) + t_0))));
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	return math.copysign(1.0, z0) * (2.0 * (math.sin(t_0) * math.sin(((math.pi * 0.5) + t_0))))

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	return Float64(copysign(1.0, z0) * Float64(2.0 * Float64(sin(t_0) * sin(Float64(Float64(pi * 0.5) + t_0)))))
end

MATLAB

function tmp = code(z0)
	t_0 = abs(z0) * pi;
	tmp = (sign(z0) * abs(1.0)) * (2.0 * (sin(t_0) * sin(((pi * 0.5) + t_0))));
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(2 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(N[(Pi * 1/2), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.9%

   \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

   14. lower-*.f64 52.9%

       \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

3. Applied rewrites 52.9%

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

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 52.8%

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

5. Applied rewrites 52.8%

   \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{1}{2} + z0 \cdot \pi\right)}\right) \]
6. Add Preprocessing

Alternative 5: 52.9% accurate, 0.5× speedup

Math

\[2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (* 2 (* (sin (* z0 PI)) (cos (* z0 PI)))))

C

double code(double z0) {
	return 2.0 * (sin((z0 * ((double) M_PI))) * cos((z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return 2.0 * (Math.sin((z0 * Math.PI)) * Math.cos((z0 * Math.PI)));
}

Python

def code(z0):
	return 2.0 * (math.sin((z0 * math.pi)) * math.cos((z0 * math.pi)))

Julia

function code(z0)
	return Float64(2.0 * Float64(sin(Float64(z0 * pi)) * cos(Float64(z0 * pi))))
end

MATLAB

function tmp = code(z0)
	tmp = 2.0 * (sin((z0 * pi)) * cos((z0 * pi)));
end

Wolfram

code[z0_] := N[(2 * N[(N[Sin[N[(z0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(z0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot z0\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(\pi \cdot z0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]

   11. lower-sin.f64 N/A

       \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(z0 \cdot \pi\right)} \cdot \cos \left(z0 \cdot \pi\right)\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \color{blue}{\cos \left(z0 \cdot \pi\right)}\right) \]

   14. lower-*.f64 52.9%

       \[\leadsto 2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

3. Applied rewrites 52.9%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(sin (* (+ PI PI) z0))"
  :precision binary64
  (sin (* (+ PI PI) z0)))