(cos (* 1/180 (* z0 PI)))

Percentage Accurate: 56.9% → 57.0%

Time: 4.0s

Alternatives: 2

Speedup: 1.0×

Specification

Math

\[\cos \left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* 1/180 (* z0 PI))))

C

double code(double z0) {
	return cos((0.005555555555555556 * (z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.cos((0.005555555555555556 * (z0 * Math.PI)));
}

Python

def code(z0):
	return math.cos((0.005555555555555556 * (z0 * math.pi)))

Julia

function code(z0)
	return cos(Float64(0.005555555555555556 * Float64(z0 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((0.005555555555555556 * (z0 * pi)));
end

Wolfram

code[z0_] := N[Cos[N[(1/180 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 56.9% accurate, 1.0× speedup

Math

\[\cos \left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* 1/180 (* z0 PI))))

C

double code(double z0) {
	return cos((0.005555555555555556 * (z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.cos((0.005555555555555556 * (z0 * Math.PI)));
}

Python

def code(z0):
	return math.cos((0.005555555555555556 * (z0 * math.pi)))

Julia

function code(z0)
	return cos(Float64(0.005555555555555556 * Float64(z0 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((0.005555555555555556 * (z0 * pi)));
end

Wolfram

code[z0_] := N[Cos[N[(1/180 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 57.0% accurate, 0.3× speedup

Math

\[\sin \left(\left(\left({\pi}^{\frac{2}{3}} \cdot \left|z0\right|\right) \cdot \frac{1}{180}\right) \cdot \left(-\sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin
 (-
  (* (* (* (pow PI 2/3) (fabs z0)) 1/180) (- (cbrt PI)))
  (* PI -1/2))))

C

double code(double z0) {
	return sin(((((pow(((double) M_PI), 0.6666666666666666) * fabs(z0)) * 0.005555555555555556) * -cbrt(((double) M_PI))) - (((double) M_PI) * -0.5)));
}

Java

public static double code(double z0) {
	return Math.sin(((((Math.pow(Math.PI, 0.6666666666666666) * Math.abs(z0)) * 0.005555555555555556) * -Math.cbrt(Math.PI)) - (Math.PI * -0.5)));
}

Julia

function code(z0)
	return sin(Float64(Float64(Float64(Float64((pi ^ 0.6666666666666666) * abs(z0)) * 0.005555555555555556) * Float64(-cbrt(pi))) - Float64(pi * -0.5)))
end

Wolfram

code[z0_] := N[Sin[N[(N[(N[(N[(N[Power[Pi, 2/3], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 1/180), $MachinePrecision] * (-N[Power[Pi, 1/3], $MachinePrecision])), $MachinePrecision] - N[(Pi * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\cos \left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. add-flip N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{180} \cdot \left(z0 \cdot \pi\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

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

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sin \left(\color{blue}{\frac{-1}{180}} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\frac{\color{blue}{\pi}}{2}\right)\right)\right) \]

   15. mult-flip N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   17. metadata-eval N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

   19. metadata-eval N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{1}{-2}}\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right) \]

   21. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right) \]

   22. metadata-eval N/A

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right) \]

   23. metadata-eval 57.0%

       \[\leadsto \sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

3. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\sin \left(\frac{-1}{180} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{-1}{2}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{-1}{180} \cdot \left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{-1}{180} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   3. metadata-eval N/A

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{-1}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \pi \cdot \frac{-1}{2}\right) \]

   5. unpow1 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \color{blue}{{\pi}^{1}}\right) - \pi \cdot \frac{-1}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot {\pi}^{\color{blue}{\left(\frac{2}{3} + \frac{1}{3}\right)}}\right) - \pi \cdot \frac{-1}{2}\right) \]

   7. pow-prod-up N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot {\pi}^{\frac{1}{3}}\right)}\right) - \pi \cdot \frac{-1}{2}\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot {\pi}^{\frac{1}{3}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   9. pow1/3 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(z0 \cdot \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   11. associate-*l* N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \color{blue}{\left(\left(z0 \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)} - \pi \cdot \frac{-1}{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z0\right)} \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z0\right)} \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

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

       \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot \sqrt[3]{\pi}\right)\right)\right)} - \pi \cdot \frac{-1}{2}\right) \]

   15. associate-*l* N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{180} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot \sqrt[3]{\pi}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{180} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right)} \cdot \sqrt[3]{\pi}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{\pi}\right)\right)} - \pi \cdot \frac{-1}{2}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{\pi}\right)\right)} - \pi \cdot \frac{-1}{2}\right) \]

5. Applied rewrites 57.0%

   \[\leadsto \sin \left(\color{blue}{\left(\left({\pi}^{\frac{2}{3}} \cdot z0\right) \cdot \frac{1}{180}\right) \cdot \left(-\sqrt[3]{\pi}\right)} - \pi \cdot \frac{-1}{2}\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(cos (* 1/180 (* z0 PI)))"
  :precision binary64
  (cos (* 1/180 (* z0 PI))))