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

Percentage Accurate: 52.9% → 65.8%

Time: 1.1min

Alternatives: 10

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\\ t_1 := \log \left({\left(e^{\pi}\right)}^{t\_0}\right)\\ t_2 := \left|z0\right| + \left|z0\right|\\ t_3 := \pi \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1120000000000000:\\ \;\;\;\;\sin \left(\left(t\_2 \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(t\_2 \cdot \pi\right), t\_0, \left(\pi + \pi\right), \left(t\_3 \cdot \pi\right), \left(\left|z0\right|\right)\right)}{t\_1 + \left(t\_1 + t\_3 \cdot \left(\left(-\pi\right) \cdot \left|z0\right|\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (* (fabs z0) (fabs z0)) PI))
       (t_1 (log (pow (exp PI) t_0)))
       (t_2 (+ (fabs z0) (fabs z0)))
       (t_3 (* PI (fabs z0))))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 1120000000000000)
     (sin (* (* t_2 (pow PI 2/3)) (pow (pow PI 1/6) 2)))
     (sin
      (/
       (304-z0z1z2z3z4 (* t_2 PI) t_0 (+ PI PI) (* t_3 PI) (fabs z0))
       (+ t_1 (+ t_1 (* t_3 (* (- PI) (fabs z0)))))))))))

Derivation

1. Split input into 2 regimes

2. if z0 < 1.12e15

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

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

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

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

   7. Applied rewrites 52.8%

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

   if 1.12e15 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.4%

      \[\leadsto \sin \left(\frac{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}}{\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. add-log-exp N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)} + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      13. log-pow-rev N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      14. lower-log.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \color{blue}{\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\color{blue}{\pi}}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      17. lower-exp.f64 16.5%

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\color{blue}{\left(e^{\pi}\right)}}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z0\right)}}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\color{blue}{\left(\pi \cdot z0\right)} \cdot z0\right)}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      20. associate-*l* N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\pi \cdot \left(z0 \cdot z0\right)\right)}}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\pi \cdot \color{blue}{\left(z0 \cdot z0\right)}\right)}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      22. *-commutative N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

      23. lift-*.f64 16.5%

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

   9. Applied rewrites 16.5%

      \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\color{blue}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right)} + \left(\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(z0 \cdot z0\right) \cdot \left(\pi \cdot \pi\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\left(z0 \cdot z0\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \pi} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)} \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)} \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\pi \cdot \left(z0 \cdot z0\right)\right)} \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\left(\pi \cdot \color{blue}{\left(z0 \cdot z0\right)}\right) \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       8. associate-*l* N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z0\right)} \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\left(\color{blue}{\left(\pi \cdot z0\right)} \cdot z0\right) \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z0\right)} \cdot \pi + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       11. lift-PI.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \color{blue}{\mathsf{PI}\left(\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       12. add-log-exp N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       13. log-pow-rev N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       14. lower-log.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       15. lower-pow.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \color{blue}{\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right)} + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       16. lift-PI.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\color{blue}{\pi}}\right)}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       17. lower-exp.f64 16.4%

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\color{blue}{\left(e^{\pi}\right)}}^{\left(\left(\pi \cdot z0\right) \cdot z0\right)}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(\pi \cdot z0\right) \cdot z0\right)}}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\left(\color{blue}{\left(\pi \cdot z0\right)} \cdot z0\right)}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       20. associate-*l* N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\pi \cdot \left(z0 \cdot z0\right)\right)}}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       21. lift-*.f64 N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\left(\pi \cdot \color{blue}{\left(z0 \cdot z0\right)}\right)}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       22. *-commutative N/A

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

       23. lift-*.f64 16.4%

           \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\log \left({\left(e^{\pi}\right)}^{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) + \left(\log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) + \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0\right)\right)}\right) \]

   11. Applied rewrites 16.4%

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

Alternative 2: 65.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|z0\right|\\ t_1 := \left(-\pi\right) \cdot \left|z0\right|\\ t_2 := \log \left({\left(e^{\pi}\right)}^{\left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right)}\right)\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1120000000000000:\\ \;\;\;\;\sin \left(\left(\left(\left|z0\right| + \left|z0\right|\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{{t\_0}^{3} - {t\_1}^{3}}{t\_2 + \left(t\_2 + t\_0 \cdot t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (fabs z0)))
       (t_1 (* (- PI) (fabs z0)))
       (t_2 (log (pow (exp PI) (* (* (fabs z0) (fabs z0)) PI)))))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 1120000000000000)
     (sin
      (*
       (* (+ (fabs z0) (fabs z0)) (pow PI 2/3))
       (pow (pow PI 1/6) 2)))
     (sin
      (/ (- (pow t_0 3) (pow t_1 3)) (+ t_2 (+ t_2 (* t_0 t_1)))))))))

C

double code(double z0) {
	double t_0 = ((double) M_PI) * fabs(z0);
	double t_1 = -((double) M_PI) * fabs(z0);
	double t_2 = log(pow(exp(((double) M_PI)), ((fabs(z0) * fabs(z0)) * ((double) M_PI))));
	double tmp;
	if (fabs(z0) <= 1.12e+15) {
		tmp = sin((((fabs(z0) + fabs(z0)) * pow(((double) M_PI), 0.6666666666666666)) * pow(pow(((double) M_PI), 0.16666666666666666), 2.0)));
	} else {
		tmp = sin(((pow(t_0, 3.0) - pow(t_1, 3.0)) / (t_2 + (t_2 + (t_0 * t_1)))));
	}
	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.PI * Math.abs(z0);
	double t_2 = Math.log(Math.pow(Math.exp(Math.PI), ((Math.abs(z0) * Math.abs(z0)) * Math.PI)));
	double tmp;
	if (Math.abs(z0) <= 1.12e+15) {
		tmp = Math.sin((((Math.abs(z0) + Math.abs(z0)) * Math.pow(Math.PI, 0.6666666666666666)) * Math.pow(Math.pow(Math.PI, 0.16666666666666666), 2.0)));
	} else {
		tmp = Math.sin(((Math.pow(t_0, 3.0) - Math.pow(t_1, 3.0)) / (t_2 + (t_2 + (t_0 * t_1)))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.pi * math.fabs(z0)
	t_1 = -math.pi * math.fabs(z0)
	t_2 = math.log(math.pow(math.exp(math.pi), ((math.fabs(z0) * math.fabs(z0)) * math.pi)))
	tmp = 0
	if math.fabs(z0) <= 1.12e+15:
		tmp = math.sin((((math.fabs(z0) + math.fabs(z0)) * math.pow(math.pi, 0.6666666666666666)) * math.pow(math.pow(math.pi, 0.16666666666666666), 2.0)))
	else:
		tmp = math.sin(((math.pow(t_0, 3.0) - math.pow(t_1, 3.0)) / (t_2 + (t_2 + (t_0 * t_1)))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(pi * abs(z0))
	t_1 = Float64(Float64(-pi) * abs(z0))
	t_2 = log((exp(pi) ^ Float64(Float64(abs(z0) * abs(z0)) * pi)))
	tmp = 0.0
	if (abs(z0) <= 1.12e+15)
		tmp = sin(Float64(Float64(Float64(abs(z0) + abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	else
		tmp = sin(Float64(Float64((t_0 ^ 3.0) - (t_1 ^ 3.0)) / Float64(t_2 + Float64(t_2 + Float64(t_0 * t_1)))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = pi * abs(z0);
	t_1 = -pi * abs(z0);
	t_2 = log((exp(pi) ^ ((abs(z0) * abs(z0)) * pi)));
	tmp = 0.0;
	if (abs(z0) <= 1.12e+15)
		tmp = sin((((abs(z0) + abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	else
		tmp = sin((((t_0 ^ 3.0) - (t_1 ^ 3.0)) / (t_2 + (t_2 + (t_0 * t_1)))));
	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[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Power[N[Exp[Pi], $MachinePrecision], N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1120000000000000], N[Sin[N[(N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 2/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[Pi, 1/6], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[Power[t$95$0, 3], $MachinePrecision] - N[Power[t$95$1, 3], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[(t$95$2 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.12e15

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

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

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

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

   7. Applied rewrites 52.8%

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

   if 1.12e15 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-PI.f64 N/A

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

      5. add-log-exp N/A

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

      6. log-pow-rev N/A

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

      7. lower-log.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-*.f64 16.5%

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

   7. Applied rewrites 16.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-PI.f64 N/A

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

      5. add-log-exp N/A

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

      6. log-pow-rev N/A

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

      7. lower-log.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-*.f64 16.4%

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

   9. Applied rewrites 16.4%

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

Alternative 3: 62.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \left|z0\right|\\ t_1 := t\_0 \cdot \pi\\ t_2 := \left(-\pi\right) \cdot \left|z0\right|\\ t_3 := \pi \cdot \left|z0\right|\\ t_4 := t\_3 \cdot \left|z0\right|\\ t_5 := t\_0 \cdot \left(\pi \cdot \pi\right)\\ t_6 := \left|z0\right| + \left|z0\right|\\ t_7 := t\_3 \cdot \pi\\ t_8 := t\_7 \cdot \pi\\ t_9 := t\_7 \cdot \left|z0\right|\\ t_10 := t\_8 \cdot t\_0\\ t_11 := t\_3 \cdot \left(t\_2 + t\_3\right)\\ t_12 := t\_11 \cdot t\_11\\ t_13 := t\_4 \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 15000000000000000:\\ \;\;\;\;\sin \left(\left(t\_6 \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 79999999999999995719222155803854345574979318317056:\\ \;\;\;\;\sin \left(\frac{\frac{\left(t\_10 - t\_10\right) \cdot \left(t\_10 + t\_10\right)}{t\_4 \cdot t\_7 - \left(\left(t\_2 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_2}}{t\_5 + \left(t\_5 + t\_3 \cdot t\_2\right)}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 36999999999999999491511714385915989040023595931478890296747033983213553844224:\\ \;\;\;\;\sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(t\_6 \cdot \pi\right), t\_1, \left(\pi + \pi\right), t\_7, \left(\left|z0\right|\right)\right)}{\frac{{t\_9}^{3} - {t\_11}^{3}}{\left(t\_8 \cdot t\_3\right) \cdot t\_0 + \left(t\_12 + t\_9 \cdot t\_11\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{{t\_3}^{3} - {t\_2}^{3}}{\frac{t\_13 \cdot t\_13 - t\_12}{t\_1 \cdot \left(\pi + \pi\right) - t\_13}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) (fabs z0)))
       (t_1 (* t_0 PI))
       (t_2 (* (- PI) (fabs z0)))
       (t_3 (* PI (fabs z0)))
       (t_4 (* t_3 (fabs z0)))
       (t_5 (* t_0 (* PI PI)))
       (t_6 (+ (fabs z0) (fabs z0)))
       (t_7 (* t_3 PI))
       (t_8 (* t_7 PI))
       (t_9 (* t_7 (fabs z0)))
       (t_10 (* t_8 t_0))
       (t_11 (* t_3 (+ t_2 t_3)))
       (t_12 (* t_11 t_11))
       (t_13 (* t_4 PI)))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 15000000000000000)
     (sin (* (* t_6 (pow PI 2/3)) (pow (pow PI 1/6) 2)))
     (if (<=
          (fabs z0)
          79999999999999995719222155803854345574979318317056)
       (sin
        (/
         (/
          (* (- t_10 t_10) (+ t_10 t_10))
          (- (* t_4 t_7) (* (* (* t_2 PI) (fabs z0)) t_2)))
         (+ t_5 (+ t_5 (* t_3 t_2)))))
       (if (<=
            (fabs z0)
            36999999999999999491511714385915989040023595931478890296747033983213553844224)
         (sin
          (/
           (304-z0z1z2z3z4 (* t_6 PI) t_1 (+ PI PI) t_7 (fabs z0))
           (/
            (- (pow t_9 3) (pow t_11 3))
            (+ (* (* t_8 t_3) t_0) (+ t_12 (* t_9 t_11))))))
         (sin
          (/
           (- (pow t_3 3) (pow t_2 3))
           (/
            (- (* t_13 t_13) t_12)
            (- (* t_1 (+ PI PI)) t_13))))))))))

Derivation

1. Split input into 4 regimes

2. if z0 < 1.5e16

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

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

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

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

   7. Applied rewrites 52.8%

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

   if 1.5e16 < z0 < 7.9999999999999996e49

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 1.0%

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

   9. Applied rewrites 5.4%

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

   if 7.9999999999999996e49 < z0 < 3.6999999999999999e76

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 14.8%

      \[\leadsto \sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot z0\right) \cdot \pi\right), z0\right)}{\color{blue}{\frac{{\left(\left(\left(\pi \cdot z0\right) \cdot \pi\right) \cdot z0\right)}^{3} - {\left(\left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0 + \pi \cdot z0\right)\right)}^{3}}{\left(\left(\left(\left(\pi \cdot z0\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(\pi \cdot z0\right)\right) \cdot \left(z0 \cdot z0\right) + \left(\left(\left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0 + \pi \cdot z0\right)\right) \cdot \left(\left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0 + \pi \cdot z0\right)\right) + \left(\left(\left(\pi \cdot z0\right) \cdot \pi\right) \cdot z0\right) \cdot \left(\left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0 + \pi \cdot z0\right)\right)\right)}}}\right) \]

   if 3.6999999999999999e76 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.2%

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

Alternative 4: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \left|z0\right|\\ t_1 := t\_0 \cdot \pi\\ t_2 := \left(-\pi\right) \cdot \left|z0\right|\\ t_3 := t\_0 \cdot \left(\pi \cdot \pi\right)\\ t_4 := \pi \cdot \left|z0\right|\\ t_5 := t\_4 \cdot \left(t\_2 + t\_4\right)\\ t_6 := t\_4 \cdot \pi\\ t_7 := t\_6 \cdot \pi\\ t_8 := \left|z0\right| + \left|z0\right|\\ t_9 := t\_7 \cdot t\_0\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 15000000000000000:\\ \;\;\;\;\sin \left(\left(t\_8 \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 40000000000000001882405379836218783566238405631466523057114838139592998125568:\\ \;\;\;\;\sin \left(\frac{\frac{\left(t\_9 - t\_9\right) \cdot \left(t\_9 + t\_9\right)}{\left(t\_4 \cdot \left|z0\right|\right) \cdot t\_6 - \left(\left(t\_2 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_2}}{t\_3 + \left(t\_3 + t\_4 \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(t\_8 \cdot \pi\right), t\_1, \left(\pi + \pi\right), t\_6, \left(\left|z0\right|\right)\right)}{\frac{\left(t\_7 \cdot t\_4\right) \cdot t\_0 - t\_5 \cdot t\_5}{t\_1 \cdot \left(\left(\pi + \pi\right) - \pi\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) (fabs z0)))
       (t_1 (* t_0 PI))
       (t_2 (* (- PI) (fabs z0)))
       (t_3 (* t_0 (* PI PI)))
       (t_4 (* PI (fabs z0)))
       (t_5 (* t_4 (+ t_2 t_4)))
       (t_6 (* t_4 PI))
       (t_7 (* t_6 PI))
       (t_8 (+ (fabs z0) (fabs z0)))
       (t_9 (* t_7 t_0)))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 15000000000000000)
     (sin (* (* t_8 (pow PI 2/3)) (pow (pow PI 1/6) 2)))
     (if (<=
          (fabs z0)
          40000000000000001882405379836218783566238405631466523057114838139592998125568)
       (sin
        (/
         (/
          (* (- t_9 t_9) (+ t_9 t_9))
          (-
           (* (* t_4 (fabs z0)) t_6)
           (* (* (* t_2 PI) (fabs z0)) t_2)))
         (+ t_3 (+ t_3 (* t_4 t_2)))))
       (sin
        (/
         (304-z0z1z2z3z4 (* t_8 PI) t_1 (+ PI PI) t_6 (fabs z0))
         (/
          (- (* (* t_7 t_4) t_0) (* t_5 t_5))
          (* t_1 (- (+ PI PI) PI))))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < 1.5e16

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

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

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

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

   7. Applied rewrites 52.8%

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

   if 1.5e16 < z0 < 4.0000000000000002e76

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 1.0%

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

   9. Applied rewrites 5.4%

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

   if 4.0000000000000002e76 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 19.2%

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

Alternative 5: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \left|z0\right|\\ t_1 := t\_0 \cdot \left(\pi \cdot \pi\right)\\ t_2 := \pi \cdot \left|z0\right|\\ t_3 := t\_2 \cdot \left|z0\right|\\ t_4 := t\_2 \cdot \pi\\ t_5 := \left(t\_4 \cdot \pi\right) \cdot t\_0\\ t_6 := \left(-\pi\right) \cdot \left|z0\right|\\ t_7 := t\_2 \cdot \left(t\_6 + t\_2\right)\\ t_8 := t\_3 \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 15000000000000000:\\ \;\;\;\;\sin \left(\left(\left(\left|z0\right| + \left|z0\right|\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot {\left({\pi}^{\frac{1}{6}}\right)}^{2}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 40000000000000001882405379836218783566238405631466523057114838139592998125568:\\ \;\;\;\;\sin \left(\frac{\frac{\left(t\_5 - t\_5\right) \cdot \left(t\_5 + t\_5\right)}{t\_3 \cdot t\_4 - \left(\left(t\_6 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_6}}{t\_1 + \left(t\_1 + t\_2 \cdot t\_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{{t\_2}^{3} - {t\_6}^{3}}{\frac{t\_8 \cdot t\_8 - t\_7 \cdot t\_7}{\left(t\_0 \cdot \pi\right) \cdot \left(\pi + \pi\right) - t\_8}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) (fabs z0)))
       (t_1 (* t_0 (* PI PI)))
       (t_2 (* PI (fabs z0)))
       (t_3 (* t_2 (fabs z0)))
       (t_4 (* t_2 PI))
       (t_5 (* (* t_4 PI) t_0))
       (t_6 (* (- PI) (fabs z0)))
       (t_7 (* t_2 (+ t_6 t_2)))
       (t_8 (* t_3 PI)))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 15000000000000000)
     (sin
      (*
       (* (+ (fabs z0) (fabs z0)) (pow PI 2/3))
       (pow (pow PI 1/6) 2)))
     (if (<=
          (fabs z0)
          40000000000000001882405379836218783566238405631466523057114838139592998125568)
       (sin
        (/
         (/
          (* (- t_5 t_5) (+ t_5 t_5))
          (- (* t_3 t_4) (* (* (* t_6 PI) (fabs z0)) t_6)))
         (+ t_1 (+ t_1 (* t_2 t_6)))))
       (sin
        (/
         (- (pow t_2 3) (pow t_6 3))
         (/
          (- (* t_8 t_8) (* t_7 t_7))
          (- (* (* t_0 PI) (+ PI PI)) t_8)))))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * fabs(z0);
	double t_1 = t_0 * (((double) M_PI) * ((double) M_PI));
	double t_2 = ((double) M_PI) * fabs(z0);
	double t_3 = t_2 * fabs(z0);
	double t_4 = t_2 * ((double) M_PI);
	double t_5 = (t_4 * ((double) M_PI)) * t_0;
	double t_6 = -((double) M_PI) * fabs(z0);
	double t_7 = t_2 * (t_6 + t_2);
	double t_8 = t_3 * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.5e+16) {
		tmp = sin((((fabs(z0) + fabs(z0)) * pow(((double) M_PI), 0.6666666666666666)) * pow(pow(((double) M_PI), 0.16666666666666666), 2.0)));
	} else if (fabs(z0) <= 4e+76) {
		tmp = sin(((((t_5 - t_5) * (t_5 + t_5)) / ((t_3 * t_4) - (((t_6 * ((double) M_PI)) * fabs(z0)) * t_6))) / (t_1 + (t_1 + (t_2 * t_6)))));
	} else {
		tmp = sin(((pow(t_2, 3.0) - pow(t_6, 3.0)) / (((t_8 * t_8) - (t_7 * t_7)) / (((t_0 * ((double) M_PI)) * (((double) M_PI) + ((double) M_PI))) - t_8))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.abs(z0);
	double t_1 = t_0 * (Math.PI * Math.PI);
	double t_2 = Math.PI * Math.abs(z0);
	double t_3 = t_2 * Math.abs(z0);
	double t_4 = t_2 * Math.PI;
	double t_5 = (t_4 * Math.PI) * t_0;
	double t_6 = -Math.PI * Math.abs(z0);
	double t_7 = t_2 * (t_6 + t_2);
	double t_8 = t_3 * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.5e+16) {
		tmp = Math.sin((((Math.abs(z0) + Math.abs(z0)) * Math.pow(Math.PI, 0.6666666666666666)) * Math.pow(Math.pow(Math.PI, 0.16666666666666666), 2.0)));
	} else if (Math.abs(z0) <= 4e+76) {
		tmp = Math.sin(((((t_5 - t_5) * (t_5 + t_5)) / ((t_3 * t_4) - (((t_6 * Math.PI) * Math.abs(z0)) * t_6))) / (t_1 + (t_1 + (t_2 * t_6)))));
	} else {
		tmp = Math.sin(((Math.pow(t_2, 3.0) - Math.pow(t_6, 3.0)) / (((t_8 * t_8) - (t_7 * t_7)) / (((t_0 * Math.PI) * (Math.PI + Math.PI)) - t_8))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.fabs(z0)
	t_1 = t_0 * (math.pi * math.pi)
	t_2 = math.pi * math.fabs(z0)
	t_3 = t_2 * math.fabs(z0)
	t_4 = t_2 * math.pi
	t_5 = (t_4 * math.pi) * t_0
	t_6 = -math.pi * math.fabs(z0)
	t_7 = t_2 * (t_6 + t_2)
	t_8 = t_3 * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.5e+16:
		tmp = math.sin((((math.fabs(z0) + math.fabs(z0)) * math.pow(math.pi, 0.6666666666666666)) * math.pow(math.pow(math.pi, 0.16666666666666666), 2.0)))
	elif math.fabs(z0) <= 4e+76:
		tmp = math.sin(((((t_5 - t_5) * (t_5 + t_5)) / ((t_3 * t_4) - (((t_6 * math.pi) * math.fabs(z0)) * t_6))) / (t_1 + (t_1 + (t_2 * t_6)))))
	else:
		tmp = math.sin(((math.pow(t_2, 3.0) - math.pow(t_6, 3.0)) / (((t_8 * t_8) - (t_7 * t_7)) / (((t_0 * math.pi) * (math.pi + math.pi)) - t_8))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) * abs(z0))
	t_1 = Float64(t_0 * Float64(pi * pi))
	t_2 = Float64(pi * abs(z0))
	t_3 = Float64(t_2 * abs(z0))
	t_4 = Float64(t_2 * pi)
	t_5 = Float64(Float64(t_4 * pi) * t_0)
	t_6 = Float64(Float64(-pi) * abs(z0))
	t_7 = Float64(t_2 * Float64(t_6 + t_2))
	t_8 = Float64(t_3 * pi)
	tmp = 0.0
	if (abs(z0) <= 1.5e+16)
		tmp = sin(Float64(Float64(Float64(abs(z0) + abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	elseif (abs(z0) <= 4e+76)
		tmp = sin(Float64(Float64(Float64(Float64(t_5 - t_5) * Float64(t_5 + t_5)) / Float64(Float64(t_3 * t_4) - Float64(Float64(Float64(t_6 * pi) * abs(z0)) * t_6))) / Float64(t_1 + Float64(t_1 + Float64(t_2 * t_6)))));
	else
		tmp = sin(Float64(Float64((t_2 ^ 3.0) - (t_6 ^ 3.0)) / Float64(Float64(Float64(t_8 * t_8) - Float64(t_7 * t_7)) / Float64(Float64(Float64(t_0 * pi) * Float64(pi + pi)) - t_8))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) * abs(z0);
	t_1 = t_0 * (pi * pi);
	t_2 = pi * abs(z0);
	t_3 = t_2 * abs(z0);
	t_4 = t_2 * pi;
	t_5 = (t_4 * pi) * t_0;
	t_6 = -pi * abs(z0);
	t_7 = t_2 * (t_6 + t_2);
	t_8 = t_3 * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.5e+16)
		tmp = sin((((abs(z0) + abs(z0)) * (pi ^ 0.6666666666666666)) * ((pi ^ 0.16666666666666666) ^ 2.0)));
	elseif (abs(z0) <= 4e+76)
		tmp = sin(((((t_5 - t_5) * (t_5 + t_5)) / ((t_3 * t_4) - (((t_6 * pi) * abs(z0)) * t_6))) / (t_1 + (t_1 + (t_2 * t_6)))));
	else
		tmp = sin((((t_2 ^ 3.0) - (t_6 ^ 3.0)) / (((t_8 * t_8) - (t_7 * t_7)) / (((t_0 * pi) * (pi + pi)) - t_8))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * Pi), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * Pi), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 * N[(t$95$6 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$3 * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 15000000000000000], N[Sin[N[(N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 2/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[Pi, 1/6], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 40000000000000001882405379836218783566238405631466523057114838139592998125568], N[Sin[N[(N[(N[(N[(t$95$5 - t$95$5), $MachinePrecision] * N[(t$95$5 + t$95$5), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * t$95$4), $MachinePrecision] - N[(N[(N[(t$95$6 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(t$95$1 + N[(t$95$2 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[Power[t$95$2, 3], $MachinePrecision] - N[Power[t$95$6, 3], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$8 * t$95$8), $MachinePrecision] - N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 1.5e16

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

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

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

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

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

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

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

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

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

      8. lower-unsound-/.f64 52.8%

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

   5. Applied rewrites 52.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 52.8%

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

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

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

      5. lower-unsound-/.f64 52.8%

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

      6. lift-/.f64 N/A

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

      7. metadata-eval 52.8%

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

   7. Applied rewrites 52.8%

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

   if 1.5e16 < z0 < 4.0000000000000002e76

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 1.0%

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

   9. Applied rewrites 5.4%

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

   if 4.0000000000000002e76 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.2%

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

Alternative 6: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(-\pi\right) \cdot \left|z0\right|\\ t_1 := \pi \cdot \left|z0\right|\\ t_2 := t\_1 \cdot \pi\\ t_3 := t\_2 \cdot \pi\\ t_4 := t\_1 \cdot \left(t\_0 + t\_1\right)\\ t_5 := \left|z0\right| \cdot \left|z0\right|\\ t_6 := t\_5 \cdot \pi\\ t_7 := t\_3 \cdot t\_5\\ t_8 := t\_5 \cdot \left(\pi \cdot \pi\right)\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 15000000000000000:\\ \;\;\;\;2 \cdot \left(\sin \left(t\_1 - \pi \cdot \frac{-1}{2}\right) \cdot \sin t\_1\right)\\ \mathbf{elif}\;\left|z0\right| \leq 40000000000000001882405379836218783566238405631466523057114838139592998125568:\\ \;\;\;\;\sin \left(\frac{\frac{\left(t\_7 - t\_7\right) \cdot \left(t\_7 + t\_7\right)}{\left(t\_1 \cdot \left|z0\right|\right) \cdot t\_2 - \left(\left(t\_0 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_0}}{t\_8 + \left(t\_8 + t\_1 \cdot t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right), t\_6, \left(\pi + \pi\right), t\_2, \left(\left|z0\right|\right)\right)}{\frac{\left(t\_3 \cdot t\_1\right) \cdot t\_5 - t\_4 \cdot t\_4}{t\_6 \cdot \left(\left(\pi + \pi\right) - \pi\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (- PI) (fabs z0)))
       (t_1 (* PI (fabs z0)))
       (t_2 (* t_1 PI))
       (t_3 (* t_2 PI))
       (t_4 (* t_1 (+ t_0 t_1)))
       (t_5 (* (fabs z0) (fabs z0)))
       (t_6 (* t_5 PI))
       (t_7 (* t_3 t_5))
       (t_8 (* t_5 (* PI PI))))
  (*
   (copysign 1 z0)
   (if (<= (fabs z0) 15000000000000000)
     (* 2 (* (sin (- t_1 (* PI -1/2))) (sin t_1)))
     (if (<=
          (fabs z0)
          40000000000000001882405379836218783566238405631466523057114838139592998125568)
       (sin
        (/
         (/
          (* (- t_7 t_7) (+ t_7 t_7))
          (-
           (* (* t_1 (fabs z0)) t_2)
           (* (* (* t_0 PI) (fabs z0)) t_0)))
         (+ t_8 (+ t_8 (* t_1 t_0)))))
       (sin
        (/
         (304-z0z1z2z3z4
          (* (+ (fabs z0) (fabs z0)) PI)
          t_6
          (+ PI PI)
          t_2
          (fabs z0))
         (/
          (- (* (* t_3 t_1) t_5) (* t_4 t_4))
          (* t_6 (- (+ PI PI) PI))))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < 1.5e16

   1. Initial program 52.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. count-2 N/A

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

      6. sin-2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 52.9%

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

   3. Applied rewrites 52.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 52.8%

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

   5. Applied rewrites 52.8%

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

   if 1.5e16 < z0 < 4.0000000000000002e76

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 1.0%

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

   9. Applied rewrites 5.4%

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

   if 4.0000000000000002e76 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 19.2%

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

Alternative 7: 54.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 \cdot z0\right) \cdot \pi\\ t_1 := \left(\pi \cdot z0\right) \cdot \pi\\ t_2 := \left(\pi \cdot z0\right) \cdot \left(\left(-\pi\right) \cdot z0 + \pi \cdot z0\right)\\ \mathbf{if}\;z0 \leq 8999999999999999938988538069254651788155375977496576:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot z0 - \pi \cdot \frac{-1}{2}\right) \cdot \sin \left(\pi \cdot z0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z0 + z0\right) \cdot \pi\right), t\_0, \left(\pi + \pi\right), t\_1, z0\right)}{\frac{\left(\left(t\_1 \cdot \pi\right) \cdot \left(\pi \cdot z0\right)\right) \cdot \left(z0 \cdot z0\right) - t\_2 \cdot t\_2}{t\_0 \cdot \left(\left(\pi + \pi\right) - \pi\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (* z0 z0) PI))
       (t_1 (* (* PI z0) PI))
       (t_2 (* (* PI z0) (+ (* (- PI) z0) (* PI z0)))))
  (if (<= z0 8999999999999999938988538069254651788155375977496576)
    (* 2 (* (sin (- (* PI z0) (* PI -1/2))) (sin (* PI z0))))
    (sin
     (/
      (304-z0z1z2z3z4 (* (+ z0 z0) PI) t_0 (+ PI PI) t_1 z0)
      (/
       (- (* (* (* t_1 PI) (* PI z0)) (* z0 z0)) (* t_2 t_2))
       (* t_0 (- (+ PI PI) PI))))))))

Derivation

1. Split input into 2 regimes

2. if z0 < 8.9999999999999999e51

   1. Initial program 52.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. count-2 N/A

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

      6. sin-2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 52.9%

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

   3. Applied rewrites 52.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 52.8%

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

   5. Applied rewrites 52.8%

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

   if 8.9999999999999999e51 < z0

   1. Initial program 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. lift-PI.f64 N/A

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

      16. pow1/3 N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 52.3%

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

   3. Applied rewrites 52.3%

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.4%

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

   9. Applied rewrites 19.2%

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

Alternative 8: 52.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. count-2 N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 52.9%

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

3. Applied rewrites 52.9%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. add-flip N/A

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

   5. lower--.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 52.8%

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

5. Applied rewrites 52.8%

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

Alternative 9: 52.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. count-2 N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 52.9%

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

3. Applied rewrites 52.9%

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

Reproduce

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