(cos (* -1/90 (* z0 PI)))

Percentage Accurate: 56.9% → 57.0%

Time: 4.2s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* -1/90 (* z0 PI))))

C

double code(double z0) {
	return cos((-0.011111111111111112 * (z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.cos((-0.011111111111111112 * (z0 * Math.PI)));
}

Python

def code(z0):
	return math.cos((-0.011111111111111112 * (z0 * math.pi)))

Julia

function code(z0)
	return cos(Float64(-0.011111111111111112 * Float64(z0 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((-0.011111111111111112 * (z0 * pi)));
end

Wolfram

code[z0_] := N[Cos[N[(-1/90 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 56.9% accurate, 1.0× speedup

Math

\[\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* -1/90 (* z0 PI))))

C

double code(double z0) {
	return cos((-0.011111111111111112 * (z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.cos((-0.011111111111111112 * (z0 * Math.PI)));
}

Python

def code(z0):
	return math.cos((-0.011111111111111112 * (z0 * math.pi)))

Julia

function code(z0)
	return cos(Float64(-0.011111111111111112 * Float64(z0 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((-0.011111111111111112 * (z0 * pi)));
end

Wolfram

code[z0_] := N[Cos[N[(-1/90 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 57.0% accurate, 0.3× speedup

Math

\[\sin \left(\left(\left({\pi}^{\frac{2}{3}} \cdot \frac{1}{90}\right) \cdot \left|z0\right|\right) \cdot \sqrt[3]{\pi} - \pi \cdot \frac{-1}{2}\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin
 (- (* (* (* (pow PI 2/3) 1/90) (fabs z0)) (cbrt PI)) (* PI -1/2))))

C

double code(double z0) {
	return sin(((((pow(((double) M_PI), 0.6666666666666666) * 0.011111111111111112) * fabs(z0)) * 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) * 0.011111111111111112) * Math.abs(z0)) * Math.cbrt(Math.PI)) - (Math.PI * -0.5)));
}

Julia

function code(z0)
	return sin(Float64(Float64(Float64(Float64((pi ^ 0.6666666666666666) * 0.011111111111111112) * abs(z0)) * cbrt(pi)) - Float64(pi * -0.5)))
end

Wolfram

code[z0_] := N[Sin[N[(N[(N[(N[(N[Power[Pi, 2/3], $MachinePrecision] * 1/90), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] - N[(Pi * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. add-flip N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

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

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{90}\right)\right) \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin \left(\color{blue}{\frac{1}{90}} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sin \left(\color{blue}{\left|\frac{-1}{90}\right|} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left|\frac{-1}{90}\right| \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \sin \left(\color{blue}{\frac{1}{90}} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   16. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\frac{\color{blue}{\pi}}{2}\right)\right)\right) \]

   17. mult-flip N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   19. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

   21. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{1}{-2}}\right) \]

   22. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right) \]

   23. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right) \]

   25. metadata-eval 57.0%

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

3. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{-1}{2}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{1}{90} \cdot \left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   3. *-commutative N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \pi \cdot \frac{-1}{2}\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \left(z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) - \pi \cdot \frac{-1}{2}\right) \]

   5. add-cube-cbrt N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \left(z0 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) - \pi \cdot \frac{-1}{2}\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \left(z0 \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   7. lift-cbrt.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \left(z0 \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right) - \pi \cdot \frac{-1}{2}\right) \]

   8. associate-*l* N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(z0 \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\pi}\right)} - \pi \cdot \frac{-1}{2}\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   10. pow1/3 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   11. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   12. pow1/3 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   13. pow-prod-up N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   14. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\left(z0 \cdot \color{blue}{{\pi}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z0\right)} \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z0\right)} \cdot \sqrt[3]{\pi}\right) - \pi \cdot \frac{-1}{2}\right) \]

   18. associate-*r* N/A

       \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot \sqrt[3]{\pi}} - \pi \cdot \frac{-1}{2}\right) \]

5. Applied rewrites 57.0%

   \[\leadsto \sin \left(\color{blue}{\left(\left({\pi}^{\frac{2}{3}} \cdot \frac{1}{90}\right) \cdot z0\right) \cdot \sqrt[3]{\pi}} - \pi \cdot \frac{-1}{2}\right) \]
6. Add Preprocessing

Alternative 2: 56.9% accurate, 1.0× speedup

Math

\[\sin \left(\pi \cdot \left(\frac{1}{90} \cdot \left|z0\right| - \frac{-1}{2}\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* PI (- (* 1/90 (fabs z0)) -1/2))))

C

double code(double z0) {
	return sin((((double) M_PI) * ((0.011111111111111112 * fabs(z0)) - -0.5)));
}

Java

public static double code(double z0) {
	return Math.sin((Math.PI * ((0.011111111111111112 * Math.abs(z0)) - -0.5)));
}

Python

def code(z0):
	return math.sin((math.pi * ((0.011111111111111112 * math.fabs(z0)) - -0.5)))

Julia

function code(z0)
	return sin(Float64(pi * Float64(Float64(0.011111111111111112 * abs(z0)) - -0.5)))
end

MATLAB

function tmp = code(z0)
	tmp = sin((pi * ((0.011111111111111112 * abs(z0)) - -0.5)));
end

Wolfram

code[z0_] := N[Sin[N[(Pi * N[(N[(1/90 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. add-flip N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

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

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{90}\right)\right) \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin \left(\color{blue}{\frac{1}{90}} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sin \left(\color{blue}{\left|\frac{-1}{90}\right|} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left|\frac{-1}{90}\right| \cdot \left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \sin \left(\color{blue}{\frac{1}{90}} \cdot \left(z0 \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

   16. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\frac{\color{blue}{\pi}}{2}\right)\right)\right) \]

   17. mult-flip N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

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

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   19. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

   21. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{1}{-2}}\right) \]

   22. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right) \]

   23. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \color{blue}{\pi \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right) \]

   25. metadata-eval 57.0%

       \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right) \]

3. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{-1}{2}\right)} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(\pi \cdot z0\right) - \pi \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{1}{90} \cdot \left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot z0\right)} - \pi \cdot \frac{-1}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(z0 \cdot \pi\right)} - \pi \cdot \frac{-1}{2}\right) \]

   5. associate-*r* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{90} \cdot z0\right) \cdot \pi} - \pi \cdot \frac{-1}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{90}\right)\right)} \cdot z0\right) \cdot \pi - \pi \cdot \frac{-1}{2}\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{90} \cdot z0\right)\right)} \cdot \pi - \pi \cdot \frac{-1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{z0 \cdot \frac{-1}{90}}\right)\right) \cdot \pi - \pi \cdot \frac{-1}{2}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{z0 \cdot \frac{-1}{90}}\right)\right) \cdot \pi - \pi \cdot \frac{-1}{2}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(z0 \cdot \frac{-1}{90}\right)\right) \cdot \pi - \color{blue}{\pi \cdot \frac{-1}{2}}\right) \]

   11. *-commutative N/A

       \[\leadsto \sin \left(\left(\mathsf{neg}\left(z0 \cdot \frac{-1}{90}\right)\right) \cdot \pi - \color{blue}{\frac{-1}{2} \cdot \pi}\right) \]

   12. distribute-rgt-out-- N/A

       \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(z0 \cdot \frac{-1}{90}\right)\right) - \frac{-1}{2}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(\left(\mathsf{neg}\left(z0 \cdot \frac{-1}{90}\right)\right) - \frac{-1}{2}\right)\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(\left(\mathsf{neg}\left(z0 \cdot \frac{-1}{90}\right)\right) - \frac{-1}{2}\right)}\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z0 \cdot \frac{-1}{90}}\right)\right) - \frac{-1}{2}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \sin \left(\pi \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{-1}{90} \cdot z0}\right)\right) - \frac{-1}{2}\right)\right) \]

   17. distribute-lft-neg-out N/A

       \[\leadsto \sin \left(\pi \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{90}\right)\right) \cdot z0} - \frac{-1}{2}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \sin \left(\pi \cdot \left(\color{blue}{\frac{1}{90}} \cdot z0 - \frac{-1}{2}\right)\right) \]

   19. lower-*.f64 57.1%

       \[\leadsto \sin \left(\pi \cdot \left(\color{blue}{\frac{1}{90} \cdot z0} - \frac{-1}{2}\right)\right) \]

5. Applied rewrites 57.1%

   \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(\frac{1}{90} \cdot z0 - \frac{-1}{2}\right)\right)} \]
6. Add Preprocessing

Alternative 3: 56.9% accurate, 1.0× speedup

Math

\[\cos \left(\left(\pi \cdot \frac{-1}{90}\right) \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (* (* PI -1/90) z0)))

C

double code(double z0) {
	return cos(((((double) M_PI) * -0.011111111111111112) * z0));
}

Java

public static double code(double z0) {
	return Math.cos(((Math.PI * -0.011111111111111112) * z0));
}

Python

def code(z0):
	return math.cos(((math.pi * -0.011111111111111112) * z0))

Julia

function code(z0)
	return cos(Float64(Float64(pi * -0.011111111111111112) * z0))
end

MATLAB

function tmp = code(z0)
	tmp = cos(((pi * -0.011111111111111112) * z0));
end

Wolfram

code[z0_] := N[Cos[N[(N[(Pi * -1/90), $MachinePrecision] * z0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\cos \left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{-1}{90} \cdot \left(z0 \cdot \pi\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\frac{-1}{90} \cdot \color{blue}{\left(z0 \cdot \pi\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \pi\right)} \]

   4. lift-PI.f64 N/A

      \[\leadsto \cos \left(\left(\frac{-1}{90} \cdot z0\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   5. add-cube-cbrt N/A

      \[\leadsto \cos \left(\left(\frac{-1}{90} \cdot z0\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto \cos \color{blue}{\left(\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \]

   7. add-cube-cbrt N/A

      \[\leadsto \cos \left(\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}\right) \]

   8. cbrt-prod N/A

      \[\leadsto \cos \left(\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)}\right) \]

   9. associate-*r* N/A

      \[\leadsto \cos \color{blue}{\left(\left(\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \cos \color{blue}{\left(\left(\left(\left(\frac{-1}{90} \cdot z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}}\right)} \]

3. Applied rewrites 56.8%

   \[\leadsto \cos \color{blue}{\left(\left(\left(\frac{-1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot {\pi}^{\frac{2}{9}}\right) \cdot {\pi}^{\frac{1}{9}}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\left(\left(\frac{-1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot {\pi}^{\frac{2}{9}}\right) \cdot {\pi}^{\frac{1}{9}}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \cos \left(\color{blue}{\left(\left(\frac{-1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot {\pi}^{\frac{2}{9}}\right)} \cdot {\pi}^{\frac{1}{9}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{-1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right) \cdot \left({\pi}^{\frac{2}{9}} \cdot {\pi}^{\frac{1}{9}}\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \cos \left(\color{blue}{\left(\frac{-1}{90} \cdot \left({\pi}^{\frac{2}{3}} \cdot z0\right)\right)} \cdot \left({\pi}^{\frac{2}{9}} \cdot {\pi}^{\frac{1}{9}}\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \cos \left(\left(\frac{-1}{90} \cdot \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z0\right)}\right) \cdot \left({\pi}^{\frac{2}{9}} \cdot {\pi}^{\frac{1}{9}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \cos \left(\color{blue}{\left(\left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right) \cdot z0\right)} \cdot \left({\pi}^{\frac{2}{9}} \cdot {\pi}^{\frac{1}{9}}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right)} \cdot \left({\pi}^{\frac{2}{9}} \cdot {\pi}^{\frac{1}{9}}\right)\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot \left(\color{blue}{{\pi}^{\frac{2}{9}}} \cdot {\pi}^{\frac{1}{9}}\right)\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot \left({\pi}^{\frac{2}{9}} \cdot \color{blue}{{\pi}^{\frac{1}{9}}}\right)\right) \]

   10. pow-prod-up N/A

       \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\pi}^{\left(\frac{2}{9} + \frac{1}{9}\right)}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot {\pi}^{\color{blue}{\frac{1}{3}}}\right) \]

   12. pow1/3 N/A

       \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \]

   13. lift-cbrt.f64 N/A

       \[\leadsto \cos \left(\left(z0 \cdot \left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \]

   14. associate-*l* N/A

       \[\leadsto \cos \color{blue}{\left(z0 \cdot \left(\left(\frac{-1}{90} \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)\right)} \]

5. Applied rewrites 56.9%

   \[\leadsto \cos \color{blue}{\left(\left(\pi \cdot \frac{-1}{90}\right) \cdot z0\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(cos (* -1/90 (* z0 PI)))"
  :precision binary64
  (cos (* -1/90 (* z0 PI))))