(sin (* z0 PI))

Percentage Accurate: 52.9% → 52.9%

Time: 2.1s

Alternatives: 2

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.9% accurate, 0.3× speedup

Math

\[\sin \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot {\left({\pi}^{\frac{1}{9}}\right)}^{3}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (* (pow PI 2/3) z0) (pow (pow PI 1/9) 3))))

C

double code(double z0) {
	return sin(((pow(((double) M_PI), 0.6666666666666666) * z0) * pow(pow(((double) M_PI), 0.1111111111111111), 3.0)));
}

Java

public static double code(double z0) {
	return Math.sin(((Math.pow(Math.PI, 0.6666666666666666) * z0) * Math.pow(Math.pow(Math.PI, 0.1111111111111111), 3.0)));
}

Python

def code(z0):
	return math.sin(((math.pow(math.pi, 0.6666666666666666) * z0) * math.pow(math.pow(math.pi, 0.1111111111111111), 3.0)))

Julia

function code(z0)
	return sin(Float64(Float64((pi ^ 0.6666666666666666) * z0) * ((pi ^ 0.1111111111111111) ^ 3.0)))
end

MATLAB

function tmp = code(z0)
	tmp = sin((((pi ^ 0.6666666666666666) * z0) * ((pi ^ 0.1111111111111111) ^ 3.0)));
end

Wolfram

code[z0_] := N[Sin[N[(N[(N[Power[Pi, 2/3], $MachinePrecision] * z0), $MachinePrecision] * N[Power[N[Power[Pi, 1/9], $MachinePrecision], 3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. pow1/3 N/A

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

   10. lift-PI.f64 N/A

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

   11. pow1/3 N/A

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

   12. pow-prod-up N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. lift-PI.f64 N/A

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

   16. lower-cbrt.f64 52.3%

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

3. Applied rewrites 52.3%

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

   1. rem-cube-cbrt N/A

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

   2. lift-cbrt.f64 N/A

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

   3. pow1/3 N/A

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

   4. cbrt-pow N/A

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

   5. metadata-eval N/A

      \[\leadsto \sin \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot {\left({\pi}^{\color{blue}{\frac{1}{9}}}\right)}^{3}\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \sin \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot {\color{blue}{\left({\pi}^{\frac{1}{9}}\right)}}^{3}\right) \]

   7. lower-pow.f64 52.9%

      \[\leadsto \sin \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot \color{blue}{{\left({\pi}^{\frac{1}{9}}\right)}^{3}}\right) \]

5. Applied rewrites 52.9%

   \[\leadsto \sin \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot \color{blue}{{\left({\pi}^{\frac{1}{9}}\right)}^{3}}\right) \]
6. Add Preprocessing

Reproduce

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