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

Percentage Accurate: 53.4% → 98.6%

Time: 2.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 28000000000000:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos \left(0.5 \cdot \pi - t\_0\right) - \cos \left(0.5 \cdot \pi + t\_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (fabs z0))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 28000000000000.0)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (*
      2.0
      (/
       (- (cos (- (* 0.5 PI) t_0)) (cos (+ (* 0.5 PI) t_0)))
       2.0))))))

C

double code(double z0) {
	double t_0 = ((double) M_PI) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 28000000000000.0) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else {
		tmp = 2.0 * ((cos(((0.5 * ((double) M_PI)) - t_0)) - cos(((0.5 * ((double) M_PI)) + t_0))) / 2.0);
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.PI * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 28000000000000.0) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else {
		tmp = 2.0 * ((Math.cos(((0.5 * Math.PI) - t_0)) - Math.cos(((0.5 * Math.PI) + t_0))) / 2.0);
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.pi * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 28000000000000.0:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	else:
		tmp = 2.0 * ((math.cos(((0.5 * math.pi) - t_0)) - math.cos(((0.5 * math.pi) + t_0))) / 2.0)
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(pi * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 28000000000000.0)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(Float64(Float64(0.5 * pi) - t_0)) - cos(Float64(Float64(0.5 * pi) + t_0))) / 2.0));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = pi * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 28000000000000.0)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = 2.0 * ((cos(((0.5 * pi) - t_0)) - cos(((0.5 * pi) + t_0))) / 2.0);
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 28000000000000.0], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[N[(N[(0.5 * Pi), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(N[(0.5 * Pi), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 2.8e13

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 2.8e13 < z0

   1. Initial program 53.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. count-2 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

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-exp N/A

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

      12. lift-PI.f64 N/A

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left({\left(e^{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{z0}\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)}}\right)\right) \]

      14. pow-neg N/A

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \color{blue}{\left(\frac{1}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}\right)}\right) \]

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

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left(\frac{1}{\color{blue}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}}\right)\right) \]

      16. lower-pow.f32 N/A

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left(\frac{1}{\color{blue}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}}\right)\right) \]

      17. pow-flip N/A

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

      18. lower-unsound-/.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. neg-log N/A

          \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{z0}}\right)\right)\right)}\right) \]

   3. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

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

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in z0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 4.8%

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

   8. Applied rewrites 4.8%

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

   9. Taylor expanded in z0 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 50.7%

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

   11. Applied rewrites 50.7%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|z0\right|\\ t_1 := \left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;\sin \left(\frac{0}{t\_0 \cdot 1 + \frac{\left(\left(-\pi\right) \cdot \left|z0\right|\right) \cdot t\_1}{t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left|z0\right| \cdot \left(t\_0 \cdot \frac{1}{t\_1}\right)\right) \cdot \left(\pi \cdot \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (fabs z0)))
       (t_1 (* (* (* (fabs z0) (fabs z0)) PI) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 6e+16)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 4.3e+153)
       (sin
        (/ 0.0 (+ (* t_0 1.0) (/ (* (* (- PI) (fabs z0)) t_1) t_1))))
       (sin
        (*
         (* (fabs z0) (* t_0 (/ 1.0 t_1)))
         (* PI (* (+ (fabs z0) (fabs z0)) PI)))))))))

C

double code(double z0) {
	double t_0 = ((double) M_PI) * fabs(z0);
	double t_1 = ((fabs(z0) * fabs(z0)) * ((double) M_PI)) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 6e+16) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 4.3e+153) {
		tmp = sin((0.0 / ((t_0 * 1.0) + (((-((double) M_PI) * fabs(z0)) * t_1) / t_1))));
	} else {
		tmp = sin(((fabs(z0) * (t_0 * (1.0 / t_1))) * (((double) M_PI) * ((fabs(z0) + fabs(z0)) * ((double) M_PI)))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.PI * Math.abs(z0);
	double t_1 = ((Math.abs(z0) * Math.abs(z0)) * Math.PI) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 6e+16) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 4.3e+153) {
		tmp = Math.sin((0.0 / ((t_0 * 1.0) + (((-Math.PI * Math.abs(z0)) * t_1) / t_1))));
	} else {
		tmp = Math.sin(((Math.abs(z0) * (t_0 * (1.0 / t_1))) * (Math.PI * ((Math.abs(z0) + Math.abs(z0)) * Math.PI))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.pi * math.fabs(z0)
	t_1 = ((math.fabs(z0) * math.fabs(z0)) * math.pi) * math.pi
	tmp = 0
	if math.fabs(z0) <= 6e+16:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 4.3e+153:
		tmp = math.sin((0.0 / ((t_0 * 1.0) + (((-math.pi * math.fabs(z0)) * t_1) / t_1))))
	else:
		tmp = math.sin(((math.fabs(z0) * (t_0 * (1.0 / t_1))) * (math.pi * ((math.fabs(z0) + math.fabs(z0)) * math.pi))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(pi * abs(z0))
	t_1 = Float64(Float64(Float64(abs(z0) * abs(z0)) * pi) * pi)
	tmp = 0.0
	if (abs(z0) <= 6e+16)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 4.3e+153)
		tmp = sin(Float64(0.0 / Float64(Float64(t_0 * 1.0) + Float64(Float64(Float64(Float64(-pi) * abs(z0)) * t_1) / t_1))));
	else
		tmp = sin(Float64(Float64(abs(z0) * Float64(t_0 * Float64(1.0 / t_1))) * Float64(pi * Float64(Float64(abs(z0) + abs(z0)) * pi))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = pi * abs(z0);
	t_1 = ((abs(z0) * abs(z0)) * pi) * pi;
	tmp = 0.0;
	if (abs(z0) <= 6e+16)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 4.3e+153)
		tmp = sin((0.0 / ((t_0 * 1.0) + (((-pi * abs(z0)) * t_1) / t_1))));
	else
		tmp = sin(((abs(z0) * (t_0 * (1.0 / t_1))) * (pi * ((abs(z0) + abs(z0)) * pi))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 6e+16], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 4.3e+153], N[Sin[N[(0.0 / N[(N[(t$95$0 * 1.0), $MachinePrecision] + N[(N[(N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[Abs[z0], $MachinePrecision] * N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 6e16

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 6e16 < z0 < 4.2999999999999998e153

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 9.5%

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

   if 4.2999999999999998e153 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 52.3%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.3%

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

   9. Applied rewrites 52.3%

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

Alternative 3: 77.3% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1200:\\ \;\;\;\;\sin \left(6.283185307179586 \cdot \left|z0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left|z0\right| \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \frac{1}{\left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right) \cdot \pi}\right)\right) \cdot \left(\pi \cdot \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 1200.0)
   (sin (* 6.283185307179586 (fabs z0)))
   (sin
    (*
     (*
      (fabs z0)
      (*
       (* PI (fabs z0))
       (/ 1.0 (* (* (* (fabs z0) (fabs z0)) PI) PI))))
     (* PI (* (+ (fabs z0) (fabs z0)) PI)))))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 1200.0) {
		tmp = sin((6.283185307179586 * fabs(z0)));
	} else {
		tmp = sin(((fabs(z0) * ((((double) M_PI) * fabs(z0)) * (1.0 / (((fabs(z0) * fabs(z0)) * ((double) M_PI)) * ((double) M_PI))))) * (((double) M_PI) * ((fabs(z0) + fabs(z0)) * ((double) M_PI)))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 1200.0) {
		tmp = Math.sin((6.283185307179586 * Math.abs(z0)));
	} else {
		tmp = Math.sin(((Math.abs(z0) * ((Math.PI * Math.abs(z0)) * (1.0 / (((Math.abs(z0) * Math.abs(z0)) * Math.PI) * Math.PI)))) * (Math.PI * ((Math.abs(z0) + Math.abs(z0)) * Math.PI))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 1200.0:
		tmp = math.sin((6.283185307179586 * math.fabs(z0)))
	else:
		tmp = math.sin(((math.fabs(z0) * ((math.pi * math.fabs(z0)) * (1.0 / (((math.fabs(z0) * math.fabs(z0)) * math.pi) * math.pi)))) * (math.pi * ((math.fabs(z0) + math.fabs(z0)) * math.pi))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 1200.0)
		tmp = sin(Float64(6.283185307179586 * abs(z0)));
	else
		tmp = sin(Float64(Float64(abs(z0) * Float64(Float64(pi * abs(z0)) * Float64(1.0 / Float64(Float64(Float64(abs(z0) * abs(z0)) * pi) * pi)))) * Float64(pi * Float64(Float64(abs(z0) + abs(z0)) * pi))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 1200.0)
		tmp = sin((6.283185307179586 * abs(z0)));
	else
		tmp = sin(((abs(z0) * ((pi * abs(z0)) * (1.0 / (((abs(z0) * abs(z0)) * pi) * pi)))) * (pi * ((abs(z0) + abs(z0)) * pi))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1200.0], N[Sin[N[(6.283185307179586 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[Abs[z0], $MachinePrecision] * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 1200

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \left(z0 + z0\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(z0 + z0\right)}\right) \]

      5. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{3982441812995697399929051632117}{1267650600228229401496703205376}} \cdot 2\right) \cdot z0\right) \]

      9. metadata-eval 53.4%

         \[\leadsto \sin \left(\color{blue}{6.283185307179586} \cdot z0\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(6.283185307179586 \cdot z0\right)} \]

   if 1200 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 52.3%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.3%

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

   9. Applied rewrites 52.3%

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

Alternative 4: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (* 2.9291837751230467 z0) 2.1450293971110255)))

C

double code(double z0) {
	return sin(((2.9291837751230467 * z0) * 2.1450293971110255));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin(((2.9291837751230467d0 * z0) * 2.1450293971110255d0))
end function

Java

public static double code(double z0) {
	return Math.sin(((2.9291837751230467 * z0) * 2.1450293971110255));
}

Python

def code(z0):
	return math.sin(((2.9291837751230467 * z0) * 2.1450293971110255))

Julia

function code(z0)
	return sin(Float64(Float64(2.9291837751230467 * z0) * 2.1450293971110255))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((2.9291837751230467 * z0) * 2.1450293971110255));
end

Wolfram

code[z0_] := N[Sin[N[(N[(2.9291837751230467 * z0), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-prod-up N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-cbrt.f64 52.9%

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

3. Applied rewrites 52.9%

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

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

   3. lower-*.f64 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   6. count-2 N/A

      \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   7. associate-*r* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   8. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   11. metadata-eval 53.4%

       \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]
8. Add Preprocessing

Alternative 5: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(6.283185307179586 \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* 6.283185307179586 z0)))

C

double code(double z0) {
	return sin((6.283185307179586 * z0));
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin((6.283185307179586d0 * z0))
end function

Java

public static double code(double z0) {
	return Math.sin((6.283185307179586 * z0));
}

Python

def code(z0):
	return math.sin((6.283185307179586 * z0))

Julia

function code(z0)
	return sin(Float64(6.283185307179586 * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sin((6.283185307179586 * z0));
end

Wolfram

code[z0_] := N[Sin[N[(6.283185307179586 * z0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-prod-up N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-cbrt.f64 52.9%

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

3. Applied rewrites 52.9%

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

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \left(z0 + z0\right)\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(z0 + z0\right)}\right) \]

   5. count-2 N/A

      \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\frac{3982441812995697399929051632117}{1267650600228229401496703205376}} \cdot 2\right) \cdot z0\right) \]

   9. metadata-eval 53.4%

      \[\leadsto \sin \left(\color{blue}{6.283185307179586} \cdot z0\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(6.283185307179586 \cdot z0\right)} \]
8. Add Preprocessing

Alternative 6: 51.1% accurate, 2.9× speedup

Math

\[z0 \cdot \left(\left(\pi + \pi\right) - 1.3333333333333333 \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 z0
 (- (+ PI PI) (* 1.3333333333333333 (* (* (* z0 z0) PI) (* PI PI))))))

C

double code(double z0) {
	return z0 * ((((double) M_PI) + ((double) M_PI)) - (1.3333333333333333 * (((z0 * z0) * ((double) M_PI)) * (((double) M_PI) * ((double) M_PI)))));
}

Java

public static double code(double z0) {
	return z0 * ((Math.PI + Math.PI) - (1.3333333333333333 * (((z0 * z0) * Math.PI) * (Math.PI * Math.PI))));
}

Python

def code(z0):
	return z0 * ((math.pi + math.pi) - (1.3333333333333333 * (((z0 * z0) * math.pi) * (math.pi * math.pi))))

Julia

function code(z0)
	return Float64(z0 * Float64(Float64(pi + pi) - Float64(1.3333333333333333 * Float64(Float64(Float64(z0 * z0) * pi) * Float64(pi * pi)))))
end

MATLAB

function tmp = code(z0)
	tmp = z0 * ((pi + pi) - (1.3333333333333333 * (((z0 * z0) * pi) * (pi * pi))));
end

Wolfram

code[z0_] := N[(z0 * N[(N[(Pi + Pi), $MachinePrecision] - N[(1.3333333333333333 * N[(N[(N[(z0 * z0), $MachinePrecision] * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

2. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

      \[\leadsto z0 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \color{blue}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-PI.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 51.1%

      \[\leadsto z0 \cdot \left(-1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right) \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{z0 \cdot \left(-1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + \color{blue}{2 \cdot \pi}\right) \]

   2. +-commutative N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi + \color{blue}{\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi + \frac{-4}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

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

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   5. lower--.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   7. count-2-rev N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \color{blue}{\left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   10. metadata-eval 51.1%

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - 1.3333333333333333 \cdot \left(\color{blue}{{z0}^{2}} \cdot {\pi}^{3}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{3}}\right)\right) \]

   12. lift-pow.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{3}}\right)\right) \]

   13. cube-mult N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \left(\color{blue}{\pi} \cdot \pi\right)\right)\right) \]

   17. lift-pow.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   20. lower-*.f64 51.1%

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - 1.3333333333333333 \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

6. Applied rewrites 51.1%

   \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{1.3333333333333333 \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)}\right) \]
7. Add Preprocessing

Alternative 7: 51.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. count-2 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

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

   9. lower-*.f64 N/A

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

   10. rem-log-exp N/A

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

   11. pow-exp N/A

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

   12. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left({\left(e^{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{z0}\right)\right) \]

   13. remove-double-neg N/A

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)}}\right)\right) \]

   14. pow-neg N/A

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \color{blue}{\left(\frac{1}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}\right)}\right) \]

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

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left(\frac{1}{\color{blue}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}}\right)\right) \]

   16. lower-pow.f32 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \log \left(\frac{1}{\color{blue}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\mathsf{neg}\left(z0\right)\right)}}}\right)\right) \]

   17. pow-flip N/A

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

   18. lower-unsound-/.f32 N/A

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

   19. lower-/.f32 N/A

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

   20. neg-log N/A

       \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot z0\right) \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{{\left(e^{\mathsf{PI}\left(\right)}\right)}^{z0}}\right)\right)\right)}\right) \]

3. Applied rewrites 53.4%

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

4. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.1%

      \[\leadsto 2 \cdot \left(z0 \cdot \pi\right) \]

6. Applied rewrites 51.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-*.f64 51.1%

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

8. Applied rewrites 51.1%

   \[\leadsto \left(z0 + z0\right) \cdot \color{blue}{\pi} \]
9. Add Preprocessing

Reproduce

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