(sin (* PI (* z1 z0)))

Percentage Accurate: 52.1% → 96.3%

Time: 1.7s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* PI (* z1 z0))))

C

double code(double z1, double z0) {
	return sin((((double) M_PI) * (z1 * z0)));
}

Java

public static double code(double z1, double z0) {
	return Math.sin((Math.PI * (z1 * z0)));
}

Python

def code(z1, z0):
	return math.sin((math.pi * (z1 * z0)))

Julia

function code(z1, z0)
	return sin(Float64(pi * Float64(z1 * z0)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((pi * (z1 * z0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(Pi * N[(z1 * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* PI (* z1 z0))))

C

double code(double z1, double z0) {
	return sin((((double) M_PI) * (z1 * z0)));
}

Java

public static double code(double z1, double z0) {
	return Math.sin((Math.PI * (z1 * z0)));
}

Python

def code(z1, z0):
	return math.sin((math.pi * (z1 * z0)))

Julia

function code(z1, z0)
	return sin(Float64(pi * Float64(z1 * z0)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((pi * (z1 * z0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(Pi * N[(z1 * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\left|z1\right| \cdot \left|z0\right|\right)\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log 1 \cdot \left|z0\right|\right)\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* PI (* (fabs z1) (fabs z0)))))
  (*
   (copysign 1.0 z1)
   (*
    (copysign 1.0 z0)
    (if (<= t_0 4e+31) (sin t_0) (sin (* (log 1.0) (fabs z0))))))))

C

double code(double z1, double z0) {
	double t_0 = ((double) M_PI) * (fabs(z1) * fabs(z0));
	double tmp;
	if (t_0 <= 4e+31) {
		tmp = sin(t_0);
	} else {
		tmp = sin((log(1.0) * fabs(z0)));
	}
	return copysign(1.0, z1) * (copysign(1.0, z0) * tmp);
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.PI * (Math.abs(z1) * Math.abs(z0));
	double tmp;
	if (t_0 <= 4e+31) {
		tmp = Math.sin(t_0);
	} else {
		tmp = Math.sin((Math.log(1.0) * Math.abs(z0)));
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z0) * tmp);
}

Python

def code(z1, z0):
	t_0 = math.pi * (math.fabs(z1) * math.fabs(z0))
	tmp = 0
	if t_0 <= 4e+31:
		tmp = math.sin(t_0)
	else:
		tmp = math.sin((math.log(1.0) * math.fabs(z0)))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z0) * tmp)

Julia

function code(z1, z0)
	t_0 = Float64(pi * Float64(abs(z1) * abs(z0)))
	tmp = 0.0
	if (t_0 <= 4e+31)
		tmp = sin(t_0);
	else
		tmp = sin(Float64(log(1.0) * abs(z0)));
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z0) * tmp))
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = pi * (abs(z1) * abs(z0));
	tmp = 0.0;
	if (t_0 <= 4e+31)
		tmp = sin(t_0);
	else
		tmp = sin((log(1.0) * abs(z0)));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z0) * abs(1.0)) * tmp);
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[z1], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 4e+31], N[Sin[t$95$0], $MachinePrecision], N[Sin[N[(N[Log[1.0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) (*.f64 z1 z0)) < 3.9999999999999999e31

   1. Initial program 52.1%

      \[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]

   if 3.9999999999999999e31 < (*.f64 (PI.f64) (*.f64 z1 z0))

   1. Initial program 52.1%

      \[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot z0\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z1 \cdot z0\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z1\right) \cdot z0\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z1\right) \cdot z0\right)} \]

      5. *-commutative N/A

         \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0\right) \]

      6. lower-*.f64 52.2%

         \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0\right) \]

   3. Applied rewrites 52.2%

      \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot \pi\right) \cdot z0\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \sin \left(\left(z1 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot z0\right) \]

      3. add-log-exp N/A

         \[\leadsto \sin \left(\left(z1 \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)}\right) \cdot z0\right) \]

      4. log-pow-rev N/A

         \[\leadsto \sin \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{z1}\right)} \cdot z0\right) \]

      5. lower-log.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{z1}\right)} \cdot z0\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \sin \left(\log \left({\left(e^{\color{blue}{\pi}}\right)}^{z1}\right) \cdot z0\right) \]

      7. pow-exp N/A

         \[\leadsto \sin \left(\log \color{blue}{\left(e^{\pi \cdot z1}\right)} \cdot z0\right) \]

      8. *-commutative N/A

         \[\leadsto \sin \left(\log \left(e^{\color{blue}{z1 \cdot \pi}}\right) \cdot z0\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sin \left(\log \left(e^{\color{blue}{z1 \cdot \pi}}\right) \cdot z0\right) \]

      10. lower-exp.f64 23.1%

          \[\leadsto \sin \left(\log \color{blue}{\left(e^{z1 \cdot \pi}\right)} \cdot z0\right) \]

   5. Applied rewrites 23.1%

      \[\leadsto \sin \left(\color{blue}{\log \left(e^{z1 \cdot \pi}\right)} \cdot z0\right) \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \sin \left(\log \color{blue}{1} \cdot z0\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 59.1%

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

Alternative 2: 50.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|z1\right|, \left|z0\right|\right)\\ t_1 := -t\_0\\ t_2 := \mathsf{min}\left(\left|z1\right|, \left|z0\right|\right)\\ \mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z0\right) \cdot \left(t\_2 \cdot \left(-0.16666666666666666 \cdot \left(\left(\left(\left(t\_0 \cdot \left(\left(\left(t\_2 \cdot t\_2\right) \cdot \pi\right) \cdot \pi\right)\right) \cdot \pi\right) \cdot t\_1\right) \cdot t\_1\right) + t\_0 \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (fmax (fabs z1) (fabs z0)))
       (t_1 (- t_0))
       (t_2 (fmin (fabs z1) (fabs z0))))
  (*
   (copysign 1.0 z1)
   (*
    (copysign 1.0 z0)
    (*
     t_2
     (+
      (*
       -0.16666666666666666
       (* (* (* (* t_0 (* (* (* t_2 t_2) PI) PI)) PI) t_1) t_1))
      (* t_0 PI)))))))

C

double code(double z1, double z0) {
	double t_0 = fmax(fabs(z1), fabs(z0));
	double t_1 = -t_0;
	double t_2 = fmin(fabs(z1), fabs(z0));
	return copysign(1.0, z1) * (copysign(1.0, z0) * (t_2 * ((-0.16666666666666666 * ((((t_0 * (((t_2 * t_2) * ((double) M_PI)) * ((double) M_PI))) * ((double) M_PI)) * t_1) * t_1)) + (t_0 * ((double) M_PI)))));
}

Java

public static double code(double z1, double z0) {
	double t_0 = fmax(Math.abs(z1), Math.abs(z0));
	double t_1 = -t_0;
	double t_2 = fmin(Math.abs(z1), Math.abs(z0));
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z0) * (t_2 * ((-0.16666666666666666 * ((((t_0 * (((t_2 * t_2) * Math.PI) * Math.PI)) * Math.PI) * t_1) * t_1)) + (t_0 * Math.PI))));
}

Python

def code(z1, z0):
	t_0 = fmax(math.fabs(z1), math.fabs(z0))
	t_1 = -t_0
	t_2 = fmin(math.fabs(z1), math.fabs(z0))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z0) * (t_2 * ((-0.16666666666666666 * ((((t_0 * (((t_2 * t_2) * math.pi) * math.pi)) * math.pi) * t_1) * t_1)) + (t_0 * math.pi))))

Julia

function code(z1, z0)
	t_0 = fmax(abs(z1), abs(z0))
	t_1 = Float64(-t_0)
	t_2 = fmin(abs(z1), abs(z0))
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z0) * Float64(t_2 * Float64(Float64(-0.16666666666666666 * Float64(Float64(Float64(Float64(t_0 * Float64(Float64(Float64(t_2 * t_2) * pi) * pi)) * pi) * t_1) * t_1)) + Float64(t_0 * pi)))))
end

MATLAB

function tmp = code(z1, z0)
	t_0 = max(abs(z1), abs(z0));
	t_1 = -t_0;
	t_2 = min(abs(z1), abs(z0));
	tmp = (sign(z1) * abs(1.0)) * ((sign(z0) * abs(1.0)) * (t_2 * ((-0.16666666666666666 * ((((t_0 * (((t_2 * t_2) * pi) * pi)) * pi) * t_1) * t_1)) + (t_0 * pi))));
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[Max[N[Abs[z1], $MachinePrecision], N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[Min[N[Abs[z1], $MachinePrecision], N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(t$95$2 * N[(N[(-0.16666666666666666 * N[(N[(N[(N[(t$95$0 * N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 52.1%

   \[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot z0\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z1 \cdot z0\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z0 \cdot z1\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right)} \]

   6. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

   7. lower-*.f64 52.2%

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

3. Applied rewrites 52.2%

   \[\leadsto \sin \color{blue}{\left(\left(z0 \cdot \pi\right) \cdot z1\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(z0 \cdot \pi\right) \cdot z1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

   3. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(z0 \cdot \left(\pi \cdot z1\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \sin \left(z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \sin \left(z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot \pi\right) \cdot z0\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0\right) \]

   8. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(z1 \cdot \left(\pi \cdot z0\right)\right)} \]

   9. rem-cube-cbrt N/A

      \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot z0\right)\right) \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{3} \cdot z0\right)\right) \]

   11. pow3 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot z0\right)\right) \]

   12. pow2 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   13. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   14. pow-cbrt N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \sin \left(z1 \cdot \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot z0\right)\right)}\right) \]

   18. *-commutative N/A

       \[\leadsto \sin \left(z1 \cdot \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(z0 \cdot \sqrt[3]{\pi}\right)}\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot {\pi}^{\frac{2}{3}}\right) \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right)} \]

   20. *-commutative N/A

       \[\leadsto \sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z1\right)} \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right) \]

   21. lift-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z1\right)} \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right) \]

5. Applied rewrites 51.8%

   \[\leadsto \sin \color{blue}{\left(\left({\pi}^{0.6666666666666666} \cdot z1\right) \cdot \left(\sqrt[3]{\pi} \cdot z0\right)\right)} \]

6. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \color{blue}{z0 \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \color{blue}{z0} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   9. lower-PI.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   11. lower-PI.f64 41.6%

       \[\leadsto z1 \cdot \left(-0.16666666666666666 \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right) \]

8. Applied rewrites 41.6%

   \[\leadsto \color{blue}{z1 \cdot \left(-0.16666666666666666 \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right) \]

   2. *-commutative N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot {z0}^{3}\right) + z0 \cdot \pi\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot {z0}^{3}\right) + z0 \cdot \pi\right) \]

   4. cube-mult N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot \left(z0 \cdot \left(z0 \cdot z0\right)\right)\right) + z0 \cdot \pi\right) \]

   5. associate-*r* N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right) + z0 \cdot \pi\right) \]

   6. sqr-neg-rev N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot z0\right) \cdot \left(\left(\mathsf{neg}\left(z0\right)\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right)\right) + z0 \cdot \pi\right) \]

   7. associate-*r* N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left(\left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot z0\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) + z0 \cdot \pi\right) \]

   8. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left(\left(\left(\left({z1}^{2} \cdot {\pi}^{3}\right) \cdot z0\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) + z0 \cdot \pi\right) \]

10. Applied rewrites 47.0%

    \[\leadsto z1 \cdot \left(-0.16666666666666666 \cdot \left(\left(\left(\left(z0 \cdot \left(\left(\left(z1 \cdot z1\right) \cdot \pi\right) \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(-z0\right)\right) \cdot \left(-z0\right)\right) + z0 \cdot \pi\right) \]
11. Add Preprocessing

Alternative 3: 50.0% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{min}\left(\left|z1\right|, z0\right) \cdot \left(\mathsf{max}\left(\left|z1\right|, z0\right) \cdot \pi\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (* (fmin (fabs z1) z0) (* (fmax (fabs z1) z0) PI))))

C

double code(double z1, double z0) {
	return copysign(1.0, z1) * (fmin(fabs(z1), z0) * (fmax(fabs(z1), z0) * ((double) M_PI)));
}

Java

public static double code(double z1, double z0) {
	return Math.copySign(1.0, z1) * (fmin(Math.abs(z1), z0) * (fmax(Math.abs(z1), z0) * Math.PI));
}

Python

def code(z1, z0):
	return math.copysign(1.0, z1) * (fmin(math.fabs(z1), z0) * (fmax(math.fabs(z1), z0) * math.pi))

Julia

function code(z1, z0)
	return Float64(copysign(1.0, z1) * Float64(fmin(abs(z1), z0) * Float64(fmax(abs(z1), z0) * pi)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = (sign(z1) * abs(1.0)) * (min(abs(z1), z0) * (max(abs(z1), z0) * pi));
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[Min[N[Abs[z1], $MachinePrecision], z0], $MachinePrecision] * N[(N[Max[N[Abs[z1], $MachinePrecision], z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\sin \left(\pi \cdot \left(z1 \cdot z0\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot z0\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z1 \cdot z0\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z0 \cdot z1\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z0\right) \cdot z1\right)} \]

   6. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

   7. lower-*.f64 52.2%

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

3. Applied rewrites 52.2%

   \[\leadsto \sin \color{blue}{\left(\left(z0 \cdot \pi\right) \cdot z1\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(z0 \cdot \pi\right) \cdot z1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \pi\right)} \cdot z1\right) \]

   3. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(z0 \cdot \left(\pi \cdot z1\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \sin \left(z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \sin \left(z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot \pi\right) \cdot z0\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0\right) \]

   8. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(z1 \cdot \left(\pi \cdot z0\right)\right)} \]

   9. rem-cube-cbrt N/A

      \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot z0\right)\right) \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{3} \cdot z0\right)\right) \]

   11. pow3 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot z0\right)\right) \]

   12. pow2 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   13. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   14. pow-cbrt N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \sin \left(z1 \cdot \left(\left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot z0\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \sin \left(z1 \cdot \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot z0\right)\right)}\right) \]

   18. *-commutative N/A

       \[\leadsto \sin \left(z1 \cdot \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(z0 \cdot \sqrt[3]{\pi}\right)}\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot {\pi}^{\frac{2}{3}}\right) \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right)} \]

   20. *-commutative N/A

       \[\leadsto \sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z1\right)} \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right) \]

   21. lift-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot z1\right)} \cdot \left(z0 \cdot \sqrt[3]{\pi}\right)\right) \]

5. Applied rewrites 51.8%

   \[\leadsto \sin \color{blue}{\left(\left({\pi}^{0.6666666666666666} \cdot z1\right) \cdot \left(\sqrt[3]{\pi} \cdot z0\right)\right)} \]

6. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \color{blue}{z0 \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \color{blue}{z0} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   9. lower-PI.f64 N/A

      \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \mathsf{PI}\left(\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto z1 \cdot \left(\frac{-1}{6} \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   11. lower-PI.f64 41.6%

       \[\leadsto z1 \cdot \left(-0.16666666666666666 \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right) \]

8. Applied rewrites 41.6%

   \[\leadsto \color{blue}{z1 \cdot \left(-0.16666666666666666 \cdot \left({z0}^{3} \cdot \left({z1}^{2} \cdot {\pi}^{3}\right)\right) + z0 \cdot \pi\right)} \]

9. Taylor expanded in z1 around 0

   \[\leadsto z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto z1 \cdot \left(z0 \cdot \mathsf{PI}\left(\right)\right) \]

    2. lower-PI.f64 50.1%

       \[\leadsto z1 \cdot \left(z0 \cdot \pi\right) \]

11. Applied rewrites 50.1%

    \[\leadsto z1 \cdot \left(z0 \cdot \color{blue}{\pi}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(sin (* PI (* z1 z0)))"
  :precision binary64
  (sin (* PI (* z1 z0))))