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

Percentage Accurate: 53.4% → 97.5%

Time: 2.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := t\_1 \cdot \left|z0\right|\\ t_3 := \left|z0\right| \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin \left(\frac{\left(\left(t\_0 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_1}{\log \left(e^{t\_3 \cdot \pi}\right) \cdot \pi}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;\sin \left(\frac{0}{\left(\left|z0\right| \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot \left|z0\right|\right) \cdot t\_2}{t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_0 \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{t\_3} \cdot 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0)))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* t_1 (fabs z0)))
       (t_3 (* (fabs z0) (fabs z0))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 85000.0)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 2e+99)
       (sin
        (/
         (* (* (* t_0 PI) (fabs z0)) t_1)
         (* (log (exp (* t_3 PI))) PI)))
       (if (<= (fabs z0) 4.3e+153)
         (sin
          (/
           0.0
           (+
            (* (* (fabs z0) PI) 1.0)
            (/ (* (* (- PI) (fabs z0)) t_2) t_2))))
         (sin
          (* t_0 (* (* PI (fabs z0)) (* (/ (fabs z0) t_3) 1.0))))))))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * fabs(z0);
	double t_2 = t_1 * fabs(z0);
	double t_3 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 2e+99) {
		tmp = sin(((((t_0 * ((double) M_PI)) * fabs(z0)) * t_1) / (log(exp((t_3 * ((double) M_PI)))) * ((double) M_PI))));
	} else if (fabs(z0) <= 4.3e+153) {
		tmp = sin((0.0 / (((fabs(z0) * ((double) M_PI)) * 1.0) + (((-((double) M_PI) * fabs(z0)) * t_2) / t_2))));
	} else {
		tmp = sin((t_0 * ((((double) M_PI) * fabs(z0)) * ((fabs(z0) / t_3) * 1.0))));
	}
	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 = (Math.PI * Math.PI) * Math.abs(z0);
	double t_2 = t_1 * Math.abs(z0);
	double t_3 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 2e+99) {
		tmp = Math.sin(((((t_0 * Math.PI) * Math.abs(z0)) * t_1) / (Math.log(Math.exp((t_3 * Math.PI))) * Math.PI)));
	} else if (Math.abs(z0) <= 4.3e+153) {
		tmp = Math.sin((0.0 / (((Math.abs(z0) * Math.PI) * 1.0) + (((-Math.PI * Math.abs(z0)) * t_2) / t_2))));
	} else {
		tmp = Math.sin((t_0 * ((Math.PI * Math.abs(z0)) * ((Math.abs(z0) / t_3) * 1.0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = (math.pi * math.pi) * math.fabs(z0)
	t_2 = t_1 * math.fabs(z0)
	t_3 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 2e+99:
		tmp = math.sin(((((t_0 * math.pi) * math.fabs(z0)) * t_1) / (math.log(math.exp((t_3 * math.pi))) * math.pi)))
	elif math.fabs(z0) <= 4.3e+153:
		tmp = math.sin((0.0 / (((math.fabs(z0) * math.pi) * 1.0) + (((-math.pi * math.fabs(z0)) * t_2) / t_2))))
	else:
		tmp = math.sin((t_0 * ((math.pi * math.fabs(z0)) * ((math.fabs(z0) / t_3) * 1.0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = Float64(Float64(pi * pi) * abs(z0))
	t_2 = Float64(t_1 * abs(z0))
	t_3 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 2e+99)
		tmp = sin(Float64(Float64(Float64(Float64(t_0 * pi) * abs(z0)) * t_1) / Float64(log(exp(Float64(t_3 * pi))) * pi)));
	elseif (abs(z0) <= 4.3e+153)
		tmp = sin(Float64(0.0 / Float64(Float64(Float64(abs(z0) * pi) * 1.0) + Float64(Float64(Float64(Float64(-pi) * abs(z0)) * t_2) / t_2))));
	else
		tmp = sin(Float64(t_0 * Float64(Float64(pi * abs(z0)) * Float64(Float64(abs(z0) / t_3) * 1.0))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = (pi * pi) * abs(z0);
	t_2 = t_1 * abs(z0);
	t_3 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 2e+99)
		tmp = sin(((((t_0 * pi) * abs(z0)) * t_1) / (log(exp((t_3 * pi))) * pi)));
	elseif (abs(z0) <= 4.3e+153)
		tmp = sin((0.0 / (((abs(z0) * pi) * 1.0) + (((-pi * abs(z0)) * t_2) / t_2))));
	else
		tmp = sin((t_0 * ((pi * abs(z0)) * ((abs(z0) / t_3) * 1.0))));
	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[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 2e+99], N[Sin[N[(N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Log[N[Exp[N[(t$95$3 * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 4.3e+153], N[Sin[N[(0.0 / N[(N[(N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision] * 1.0), $MachinePrecision] + N[(N[(N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(t$95$0 * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / t$95$3), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < 85000

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 85000 < z0 < 1.9999999999999999e99

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-log-exp N/A

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

      4. log-pow-rev N/A

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

      5. lower-log.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow-exp N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-exp.f64 15.0%

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

   9. Applied rewrites 15.0%

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

   if 1.9999999999999999e99 < z0 < 4.2999999999999998e153

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 9.5%

      \[\leadsto \sin \color{blue}{\left(\frac{0}{\left(z0 \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot z0\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0\right)}{\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0}}\right)} \]

   if 4.2999999999999998e153 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := t\_1 \cdot \left|z0\right|\\ t_3 := \sin \left(\frac{0}{\left(\left|z0\right| \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot \left|z0\right|\right) \cdot t\_2}{t\_2}}\right)\\ t_4 := \left|z0\right| \cdot \left|z0\right|\\ t_5 := \left(t\_4 \cdot \pi\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.96 \cdot 10^{+15}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 1.3 \cdot 10^{+74}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|z0\right| \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin \left(\frac{\left(\left(t\_0 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_1}{\sqrt{t\_5 \cdot t\_5}}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_0 \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{t\_4} \cdot 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0)))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* t_1 (fabs z0)))
       (t_3
        (sin
         (/
          0.0
          (+
           (* (* (fabs z0) PI) 1.0)
           (/ (* (* (- PI) (fabs z0)) t_2) t_2)))))
       (t_4 (* (fabs z0) (fabs z0)))
       (t_5 (* (* t_4 PI) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.96e+15)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 1.3e+74)
       t_3
       (if (<= (fabs z0) 2e+99)
         (sin (/ (* (* (* t_0 PI) (fabs z0)) t_1) (sqrt (* t_5 t_5))))
         (if (<= (fabs z0) 4.3e+153)
           t_3
           (sin
            (*
             t_0
             (* (* PI (fabs z0)) (* (/ (fabs z0) t_4) 1.0)))))))))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * fabs(z0);
	double t_2 = t_1 * fabs(z0);
	double t_3 = sin((0.0 / (((fabs(z0) * ((double) M_PI)) * 1.0) + (((-((double) M_PI) * fabs(z0)) * t_2) / t_2))));
	double t_4 = fabs(z0) * fabs(z0);
	double t_5 = (t_4 * ((double) M_PI)) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.96e+15) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (fabs(z0) <= 2e+99) {
		tmp = sin(((((t_0 * ((double) M_PI)) * fabs(z0)) * t_1) / sqrt((t_5 * t_5))));
	} else if (fabs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = sin((t_0 * ((((double) M_PI) * fabs(z0)) * ((fabs(z0) / t_4) * 1.0))));
	}
	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 = (Math.PI * Math.PI) * Math.abs(z0);
	double t_2 = t_1 * Math.abs(z0);
	double t_3 = Math.sin((0.0 / (((Math.abs(z0) * Math.PI) * 1.0) + (((-Math.PI * Math.abs(z0)) * t_2) / t_2))));
	double t_4 = Math.abs(z0) * Math.abs(z0);
	double t_5 = (t_4 * Math.PI) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.96e+15) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (Math.abs(z0) <= 2e+99) {
		tmp = Math.sin(((((t_0 * Math.PI) * Math.abs(z0)) * t_1) / Math.sqrt((t_5 * t_5))));
	} else if (Math.abs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = Math.sin((t_0 * ((Math.PI * Math.abs(z0)) * ((Math.abs(z0) / t_4) * 1.0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = (math.pi * math.pi) * math.fabs(z0)
	t_2 = t_1 * math.fabs(z0)
	t_3 = math.sin((0.0 / (((math.fabs(z0) * math.pi) * 1.0) + (((-math.pi * math.fabs(z0)) * t_2) / t_2))))
	t_4 = math.fabs(z0) * math.fabs(z0)
	t_5 = (t_4 * math.pi) * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.96e+15:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 1.3e+74:
		tmp = t_3
	elif math.fabs(z0) <= 2e+99:
		tmp = math.sin(((((t_0 * math.pi) * math.fabs(z0)) * t_1) / math.sqrt((t_5 * t_5))))
	elif math.fabs(z0) <= 4.3e+153:
		tmp = t_3
	else:
		tmp = math.sin((t_0 * ((math.pi * math.fabs(z0)) * ((math.fabs(z0) / t_4) * 1.0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = Float64(Float64(pi * pi) * abs(z0))
	t_2 = Float64(t_1 * abs(z0))
	t_3 = sin(Float64(0.0 / Float64(Float64(Float64(abs(z0) * pi) * 1.0) + Float64(Float64(Float64(Float64(-pi) * abs(z0)) * t_2) / t_2))))
	t_4 = Float64(abs(z0) * abs(z0))
	t_5 = Float64(Float64(t_4 * pi) * pi)
	tmp = 0.0
	if (abs(z0) <= 1.96e+15)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin(Float64(Float64(Float64(Float64(t_0 * pi) * abs(z0)) * t_1) / sqrt(Float64(t_5 * t_5))));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin(Float64(t_0 * Float64(Float64(pi * abs(z0)) * Float64(Float64(abs(z0) / t_4) * 1.0))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = (pi * pi) * abs(z0);
	t_2 = t_1 * abs(z0);
	t_3 = sin((0.0 / (((abs(z0) * pi) * 1.0) + (((-pi * abs(z0)) * t_2) / t_2))));
	t_4 = abs(z0) * abs(z0);
	t_5 = (t_4 * pi) * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.96e+15)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin(((((t_0 * pi) * abs(z0)) * t_1) / sqrt((t_5 * t_5))));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin((t_0 * ((pi * abs(z0)) * ((abs(z0) / t_4) * 1.0))));
	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[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.0 / N[(N[(N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision] * 1.0), $MachinePrecision] + N[(N[(N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1.96e+15], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 1.3e+74], t$95$3, If[LessEqual[N[Abs[z0], $MachinePrecision], 2e+99], N[Sin[N[(N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(t$95$5 * t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 4.3e+153], t$95$3, N[Sin[N[(t$95$0 * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < 1.96e15

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 1.96e15 < z0 < 1.3e74 or 1.9999999999999999e99 < z0 < 4.2999999999999998e153

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 9.5%

      \[\leadsto \sin \color{blue}{\left(\frac{0}{\left(z0 \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot z0\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0\right)}{\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0}}\right)} \]

   if 1.3e74 < z0 < 1.9999999999999999e99

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. fabs-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-*.f32 N/A

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

      14. lower-unsound-*.f32 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-unsound-*.f64 19.0%

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

   9. Applied rewrites 19.0%

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

   if 4.2999999999999998e153 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ t_1 := \left|z0\right| + \left|z0\right|\\ t_2 := \left(\left(\pi \cdot \pi\right) \cdot \left|z0\right|\right) \cdot \left|z0\right|\\ t_3 := \sin \left(\frac{0}{t\_0 \cdot 1 + \frac{\left(\left(-\pi\right) \cdot \left|z0\right|\right) \cdot t\_2}{t\_2}}\right)\\ t_4 := \left|z0\right| \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.96 \cdot 10^{+15}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 1.3 \cdot 10^{+74}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|z0\right| \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin \left(\frac{-\left(\left(\left(t\_0 \cdot \pi\right) \cdot t\_1\right) \cdot t\_0\right) \cdot 1}{\left(-\pi\right) \cdot \left(\sqrt{t\_4 \cdot t\_4} \cdot \pi\right)}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_1 \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{t\_4} \cdot 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI))
       (t_1 (+ (fabs z0) (fabs z0)))
       (t_2 (* (* (* PI PI) (fabs z0)) (fabs z0)))
       (t_3
        (sin
         (/
          0.0
          (+ (* t_0 1.0) (/ (* (* (- PI) (fabs z0)) t_2) t_2)))))
       (t_4 (* (fabs z0) (fabs z0))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.96e+15)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 1.3e+74)
       t_3
       (if (<= (fabs z0) 2e+99)
         (sin
          (/
           (- (* (* (* (* t_0 PI) t_1) t_0) 1.0))
           (* (- PI) (* (sqrt (* t_4 t_4)) PI))))
         (if (<= (fabs z0) 4.3e+153)
           t_3
           (sin
            (*
             t_1
             (* (* PI (fabs z0)) (* (/ (fabs z0) t_4) 1.0)))))))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	double t_1 = fabs(z0) + fabs(z0);
	double t_2 = ((((double) M_PI) * ((double) M_PI)) * fabs(z0)) * fabs(z0);
	double t_3 = sin((0.0 / ((t_0 * 1.0) + (((-((double) M_PI) * fabs(z0)) * t_2) / t_2))));
	double t_4 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 1.96e+15) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (fabs(z0) <= 2e+99) {
		tmp = sin((-((((t_0 * ((double) M_PI)) * t_1) * t_0) * 1.0) / (-((double) M_PI) * (sqrt((t_4 * t_4)) * ((double) M_PI)))));
	} else if (fabs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = sin((t_1 * ((((double) M_PI) * fabs(z0)) * ((fabs(z0) / t_4) * 1.0))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	double t_1 = Math.abs(z0) + Math.abs(z0);
	double t_2 = ((Math.PI * Math.PI) * Math.abs(z0)) * Math.abs(z0);
	double t_3 = Math.sin((0.0 / ((t_0 * 1.0) + (((-Math.PI * Math.abs(z0)) * t_2) / t_2))));
	double t_4 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 1.96e+15) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (Math.abs(z0) <= 2e+99) {
		tmp = Math.sin((-((((t_0 * Math.PI) * t_1) * t_0) * 1.0) / (-Math.PI * (Math.sqrt((t_4 * t_4)) * Math.PI))));
	} else if (Math.abs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = Math.sin((t_1 * ((Math.PI * Math.abs(z0)) * ((Math.abs(z0) / t_4) * 1.0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	t_1 = math.fabs(z0) + math.fabs(z0)
	t_2 = ((math.pi * math.pi) * math.fabs(z0)) * math.fabs(z0)
	t_3 = math.sin((0.0 / ((t_0 * 1.0) + (((-math.pi * math.fabs(z0)) * t_2) / t_2))))
	t_4 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 1.96e+15:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 1.3e+74:
		tmp = t_3
	elif math.fabs(z0) <= 2e+99:
		tmp = math.sin((-((((t_0 * math.pi) * t_1) * t_0) * 1.0) / (-math.pi * (math.sqrt((t_4 * t_4)) * math.pi))))
	elif math.fabs(z0) <= 4.3e+153:
		tmp = t_3
	else:
		tmp = math.sin((t_1 * ((math.pi * math.fabs(z0)) * ((math.fabs(z0) / t_4) * 1.0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	t_1 = Float64(abs(z0) + abs(z0))
	t_2 = Float64(Float64(Float64(pi * pi) * abs(z0)) * abs(z0))
	t_3 = sin(Float64(0.0 / Float64(Float64(t_0 * 1.0) + Float64(Float64(Float64(Float64(-pi) * abs(z0)) * t_2) / t_2))))
	t_4 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 1.96e+15)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin(Float64(Float64(-Float64(Float64(Float64(Float64(t_0 * pi) * t_1) * t_0) * 1.0)) / Float64(Float64(-pi) * Float64(sqrt(Float64(t_4 * t_4)) * pi))));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin(Float64(t_1 * Float64(Float64(pi * abs(z0)) * Float64(Float64(abs(z0) / t_4) * 1.0))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) * pi;
	t_1 = abs(z0) + abs(z0);
	t_2 = ((pi * pi) * abs(z0)) * abs(z0);
	t_3 = sin((0.0 / ((t_0 * 1.0) + (((-pi * abs(z0)) * t_2) / t_2))));
	t_4 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 1.96e+15)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin((-((((t_0 * pi) * t_1) * t_0) * 1.0) / (-pi * (sqrt((t_4 * t_4)) * pi))));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin((t_1 * ((pi * abs(z0)) * ((abs(z0) / t_4) * 1.0))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.0 / N[(N[(t$95$0 * 1.0), $MachinePrecision] + N[(N[(N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1.96e+15], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 1.3e+74], t$95$3, If[LessEqual[N[Abs[z0], $MachinePrecision], 2e+99], N[Sin[N[((-N[(N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]) / N[((-Pi) * N[(N[Sqrt[N[(t$95$4 * t$95$4), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 4.3e+153], t$95$3, N[Sin[N[(t$95$1 * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < 1.96e15

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 1.96e15 < z0 < 1.3e74 or 1.9999999999999999e99 < z0 < 4.2999999999999998e153

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 9.5%

      \[\leadsto \sin \color{blue}{\left(\frac{0}{\left(z0 \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot z0\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0\right)}{\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0}}\right)} \]

   if 1.3e74 < z0 < 1.9999999999999999e99

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 18.8%

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

   7. Applied rewrites 18.8%

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

   if 4.2999999999999998e153 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := t\_1 \cdot \left|z0\right|\\ t_3 := \sin \left(\frac{0}{\left(\left|z0\right| \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot \left|z0\right|\right) \cdot t\_2}{t\_2}}\right)\\ t_4 := \left|z0\right| \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.96 \cdot 10^{+15}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 1.3 \cdot 10^{+74}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|z0\right| \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin \left(\frac{\left(\left(t\_0 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_1}{\left(\sqrt{t\_4 \cdot t\_4} \cdot \pi\right) \cdot \pi}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 4.3 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin \left(t\_0 \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{t\_4} \cdot 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0)))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* t_1 (fabs z0)))
       (t_3
        (sin
         (/
          0.0
          (+
           (* (* (fabs z0) PI) 1.0)
           (/ (* (* (- PI) (fabs z0)) t_2) t_2)))))
       (t_4 (* (fabs z0) (fabs z0))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.96e+15)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 1.3e+74)
       t_3
       (if (<= (fabs z0) 2e+99)
         (sin
          (/
           (* (* (* t_0 PI) (fabs z0)) t_1)
           (* (* (sqrt (* t_4 t_4)) PI) PI)))
         (if (<= (fabs z0) 4.3e+153)
           t_3
           (sin
            (*
             t_0
             (* (* PI (fabs z0)) (* (/ (fabs z0) t_4) 1.0)))))))))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * fabs(z0);
	double t_2 = t_1 * fabs(z0);
	double t_3 = sin((0.0 / (((fabs(z0) * ((double) M_PI)) * 1.0) + (((-((double) M_PI) * fabs(z0)) * t_2) / t_2))));
	double t_4 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 1.96e+15) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (fabs(z0) <= 2e+99) {
		tmp = sin(((((t_0 * ((double) M_PI)) * fabs(z0)) * t_1) / ((sqrt((t_4 * t_4)) * ((double) M_PI)) * ((double) M_PI))));
	} else if (fabs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = sin((t_0 * ((((double) M_PI) * fabs(z0)) * ((fabs(z0) / t_4) * 1.0))));
	}
	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 = (Math.PI * Math.PI) * Math.abs(z0);
	double t_2 = t_1 * Math.abs(z0);
	double t_3 = Math.sin((0.0 / (((Math.abs(z0) * Math.PI) * 1.0) + (((-Math.PI * Math.abs(z0)) * t_2) / t_2))));
	double t_4 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 1.96e+15) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 1.3e+74) {
		tmp = t_3;
	} else if (Math.abs(z0) <= 2e+99) {
		tmp = Math.sin(((((t_0 * Math.PI) * Math.abs(z0)) * t_1) / ((Math.sqrt((t_4 * t_4)) * Math.PI) * Math.PI)));
	} else if (Math.abs(z0) <= 4.3e+153) {
		tmp = t_3;
	} else {
		tmp = Math.sin((t_0 * ((Math.PI * Math.abs(z0)) * ((Math.abs(z0) / t_4) * 1.0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = (math.pi * math.pi) * math.fabs(z0)
	t_2 = t_1 * math.fabs(z0)
	t_3 = math.sin((0.0 / (((math.fabs(z0) * math.pi) * 1.0) + (((-math.pi * math.fabs(z0)) * t_2) / t_2))))
	t_4 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 1.96e+15:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 1.3e+74:
		tmp = t_3
	elif math.fabs(z0) <= 2e+99:
		tmp = math.sin(((((t_0 * math.pi) * math.fabs(z0)) * t_1) / ((math.sqrt((t_4 * t_4)) * math.pi) * math.pi)))
	elif math.fabs(z0) <= 4.3e+153:
		tmp = t_3
	else:
		tmp = math.sin((t_0 * ((math.pi * math.fabs(z0)) * ((math.fabs(z0) / t_4) * 1.0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = Float64(Float64(pi * pi) * abs(z0))
	t_2 = Float64(t_1 * abs(z0))
	t_3 = sin(Float64(0.0 / Float64(Float64(Float64(abs(z0) * pi) * 1.0) + Float64(Float64(Float64(Float64(-pi) * abs(z0)) * t_2) / t_2))))
	t_4 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 1.96e+15)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin(Float64(Float64(Float64(Float64(t_0 * pi) * abs(z0)) * t_1) / Float64(Float64(sqrt(Float64(t_4 * t_4)) * pi) * pi)));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin(Float64(t_0 * Float64(Float64(pi * abs(z0)) * Float64(Float64(abs(z0) / t_4) * 1.0))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = (pi * pi) * abs(z0);
	t_2 = t_1 * abs(z0);
	t_3 = sin((0.0 / (((abs(z0) * pi) * 1.0) + (((-pi * abs(z0)) * t_2) / t_2))));
	t_4 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 1.96e+15)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.3e+74)
		tmp = t_3;
	elseif (abs(z0) <= 2e+99)
		tmp = sin(((((t_0 * pi) * abs(z0)) * t_1) / ((sqrt((t_4 * t_4)) * pi) * pi)));
	elseif (abs(z0) <= 4.3e+153)
		tmp = t_3;
	else
		tmp = sin((t_0 * ((pi * abs(z0)) * ((abs(z0) / t_4) * 1.0))));
	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[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.0 / N[(N[(N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision] * 1.0), $MachinePrecision] + N[(N[(N[((-Pi) * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1.96e+15], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 1.3e+74], t$95$3, If[LessEqual[N[Abs[z0], $MachinePrecision], 2e+99], N[Sin[N[(N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(N[Sqrt[N[(t$95$4 * t$95$4), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 4.3e+153], t$95$3, N[Sin[N[(t$95$0 * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z0 < 1.96e15

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 1.96e15 < z0 < 1.3e74 or 1.9999999999999999e99 < z0 < 4.2999999999999998e153

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 9.5%

      \[\leadsto \sin \color{blue}{\left(\frac{0}{\left(z0 \cdot \pi\right) \cdot 1 + \frac{\left(\left(-\pi\right) \cdot z0\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0\right)}{\left(\left(\pi \cdot \pi\right) \cdot z0\right) \cdot z0}}\right)} \]

   if 1.3e74 < z0 < 1.9999999999999999e99

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 18.8%

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

   9. Applied rewrites 18.8%

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

   if 4.2999999999999998e153 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 5: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := \left|z0\right| \cdot \left|z0\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 10^{+31}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{elif}\;\left|z0\right| \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;\sin \left(\frac{\left(\left(t\_0 \cdot \pi\right) \cdot \left|z0\right|\right) \cdot t\_1}{\left(\sqrt{t\_2 \cdot t\_2} \cdot \pi\right) \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\left(t\_0 \cdot \left(\frac{\left|z0\right|}{t\_2} \cdot 1\right)\right) \cdot t\_1}{\pi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0)))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* (fabs z0) (fabs z0))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1e+31)
     (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
     (if (<= (fabs z0) 1.45e+102)
       (sin
        (/
         (* (* (* t_0 PI) (fabs z0)) t_1)
         (* (* (sqrt (* t_2 t_2)) PI) PI)))
       (sin (/ (* (* t_0 (* (/ (fabs z0) t_2) 1.0)) t_1) PI)))))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * fabs(z0);
	double t_2 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 1e+31) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else if (fabs(z0) <= 1.45e+102) {
		tmp = sin(((((t_0 * ((double) M_PI)) * fabs(z0)) * t_1) / ((sqrt((t_2 * t_2)) * ((double) M_PI)) * ((double) M_PI))));
	} else {
		tmp = sin((((t_0 * ((fabs(z0) / t_2) * 1.0)) * t_1) / ((double) M_PI)));
	}
	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 = (Math.PI * Math.PI) * Math.abs(z0);
	double t_2 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 1e+31) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else if (Math.abs(z0) <= 1.45e+102) {
		tmp = Math.sin(((((t_0 * Math.PI) * Math.abs(z0)) * t_1) / ((Math.sqrt((t_2 * t_2)) * Math.PI) * Math.PI)));
	} else {
		tmp = Math.sin((((t_0 * ((Math.abs(z0) / t_2) * 1.0)) * t_1) / Math.PI));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = (math.pi * math.pi) * math.fabs(z0)
	t_2 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 1e+31:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	elif math.fabs(z0) <= 1.45e+102:
		tmp = math.sin(((((t_0 * math.pi) * math.fabs(z0)) * t_1) / ((math.sqrt((t_2 * t_2)) * math.pi) * math.pi)))
	else:
		tmp = math.sin((((t_0 * ((math.fabs(z0) / t_2) * 1.0)) * t_1) / math.pi))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = Float64(Float64(pi * pi) * abs(z0))
	t_2 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 1e+31)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.45e+102)
		tmp = sin(Float64(Float64(Float64(Float64(t_0 * pi) * abs(z0)) * t_1) / Float64(Float64(sqrt(Float64(t_2 * t_2)) * pi) * pi)));
	else
		tmp = sin(Float64(Float64(Float64(t_0 * Float64(Float64(abs(z0) / t_2) * 1.0)) * t_1) / pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = (pi * pi) * abs(z0);
	t_2 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 1e+31)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	elseif (abs(z0) <= 1.45e+102)
		tmp = sin(((((t_0 * pi) * abs(z0)) * t_1) / ((sqrt((t_2 * t_2)) * pi) * pi)));
	else
		tmp = sin((((t_0 * ((abs(z0) / t_2) * 1.0)) * t_1) / pi));
	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[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 1e+31], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 1.45e+102], N[Sin[N[(N[(N[(N[(t$95$0 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[(t$95$0 * N[(N[(N[Abs[z0], $MachinePrecision] / t$95$2), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 9.9999999999999996e30

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 9.9999999999999996e30 < z0 < 1.4500000000000001e102

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 18.8%

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

   9. Applied rewrites 18.8%

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

   if 1.4500000000000001e102 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 6: 77.3% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 5.5 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{\left|z0\right| \cdot \left|z0\right|} \cdot 1\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left|z0\right|\right)}{\pi}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 5.5e+105)
   (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
   (sin
    (/
     (*
      (*
       (+ (fabs z0) (fabs z0))
       (* (/ (fabs z0) (* (fabs z0) (fabs z0))) 1.0))
      (* (* PI PI) (fabs z0)))
     PI)))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 5.5e+105) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else {
		tmp = sin(((((fabs(z0) + fabs(z0)) * ((fabs(z0) / (fabs(z0) * fabs(z0))) * 1.0)) * ((((double) M_PI) * ((double) M_PI)) * fabs(z0))) / ((double) M_PI)));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 5.5e+105) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else {
		tmp = Math.sin(((((Math.abs(z0) + Math.abs(z0)) * ((Math.abs(z0) / (Math.abs(z0) * Math.abs(z0))) * 1.0)) * ((Math.PI * Math.PI) * Math.abs(z0))) / Math.PI));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 5.5e+105:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	else:
		tmp = math.sin(((((math.fabs(z0) + math.fabs(z0)) * ((math.fabs(z0) / (math.fabs(z0) * math.fabs(z0))) * 1.0)) * ((math.pi * math.pi) * math.fabs(z0))) / math.pi))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 5.5e+105)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = sin(Float64(Float64(Float64(Float64(abs(z0) + abs(z0)) * Float64(Float64(abs(z0) / Float64(abs(z0) * abs(z0))) * 1.0)) * Float64(Float64(pi * pi) * abs(z0))) / pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 5.5e+105)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = sin(((((abs(z0) + abs(z0)) * ((abs(z0) / (abs(z0) * abs(z0))) * 1.0)) * ((pi * pi) * abs(z0))) / pi));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 5.5e+105], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 5.4999999999999998e105

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 5.4999999999999998e105 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 7: 77.3% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\sin \left(\left(2.9291837751230467 \cdot \left|z0\right|\right) \cdot 2.1450293971110255\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left(\frac{\left|z0\right|}{\left|z0\right| \cdot \left|z0\right|} \cdot 1\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 2e+126)
   (sin (* (* 2.9291837751230467 (fabs z0)) 2.1450293971110255))
   (sin
    (*
     (+ (fabs z0) (fabs z0))
     (*
      (* PI (fabs z0))
      (* (/ (fabs z0) (* (fabs z0) (fabs z0))) 1.0)))))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 2e+126) {
		tmp = sin(((2.9291837751230467 * fabs(z0)) * 2.1450293971110255));
	} else {
		tmp = sin(((fabs(z0) + fabs(z0)) * ((((double) M_PI) * fabs(z0)) * ((fabs(z0) / (fabs(z0) * fabs(z0))) * 1.0))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 2e+126) {
		tmp = Math.sin(((2.9291837751230467 * Math.abs(z0)) * 2.1450293971110255));
	} else {
		tmp = Math.sin(((Math.abs(z0) + Math.abs(z0)) * ((Math.PI * Math.abs(z0)) * ((Math.abs(z0) / (Math.abs(z0) * Math.abs(z0))) * 1.0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 2e+126:
		tmp = math.sin(((2.9291837751230467 * math.fabs(z0)) * 2.1450293971110255))
	else:
		tmp = math.sin(((math.fabs(z0) + math.fabs(z0)) * ((math.pi * math.fabs(z0)) * ((math.fabs(z0) / (math.fabs(z0) * math.fabs(z0))) * 1.0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 2e+126)
		tmp = sin(Float64(Float64(2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = sin(Float64(Float64(abs(z0) + abs(z0)) * Float64(Float64(pi * abs(z0)) * Float64(Float64(abs(z0) / Float64(abs(z0) * abs(z0))) * 1.0))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 2e+126)
		tmp = sin(((2.9291837751230467 * abs(z0)) * 2.1450293971110255));
	else
		tmp = sin(((abs(z0) + abs(z0)) * ((pi * abs(z0)) * ((abs(z0) / (abs(z0) * abs(z0))) * 1.0))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 2e+126], N[Sin[N[(N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[z0], $MachinePrecision] / N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 1.9999999999999998e126

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      6. count-2 N/A

         \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

      11. metadata-eval 53.4%

          \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

   7. Applied rewrites 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]

   if 1.9999999999999998e126 < z0

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-PI.f64 N/A

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

      10. pow1/3 N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 19.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 53.0%

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

Alternative 8: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (* 2.9291837751230467 z0) 2.1450293971110255)))

C

double code(double z0) {
	return sin(((2.9291837751230467 * z0) * 2.1450293971110255));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin(((2.9291837751230467d0 * z0) * 2.1450293971110255d0))
end function

Java

public static double code(double z0) {
	return Math.sin(((2.9291837751230467 * z0) * 2.1450293971110255));
}

Python

def code(z0):
	return math.sin(((2.9291837751230467 * z0) * 2.1450293971110255))

Julia

function code(z0)
	return sin(Float64(Float64(2.9291837751230467 * z0) * 2.1450293971110255))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((2.9291837751230467 * z0) * 2.1450293971110255));
end

Wolfram

code[z0_] := N[Sin[N[(N[(2.9291837751230467 * z0), $MachinePrecision] * 2.1450293971110255), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. pow1/3 N/A

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

   9. lift-PI.f64 N/A

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

   10. pow1/3 N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lower-cbrt.f64 52.9%

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

3. Applied rewrites 52.9%

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

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

   3. lower-*.f64 53.4%

      \[\leadsto \sin \color{blue}{\left(\left(1.4645918875615234 \cdot \left(z0 + z0\right)\right) \cdot 2.1450293971110255\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(z0 + z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   6. count-2 N/A

      \[\leadsto \sin \left(\left(\frac{824491934883991}{562949953421312} \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   7. associate-*r* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot 2\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   8. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\frac{824491934883991}{281474976710656}} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right)} \cdot z0\right) \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} + \frac{824491934883991}{562949953421312}\right) \cdot z0\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   11. metadata-eval 53.4%

       \[\leadsto \sin \left(\left(\color{blue}{2.9291837751230467} \cdot z0\right) \cdot 2.1450293971110255\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(\left(2.9291837751230467 \cdot z0\right) \cdot 2.1450293971110255\right)} \]
8. Add Preprocessing

Alternative 9: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(6.283185307179586 \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* 6.283185307179586 z0)))

C

double code(double z0) {
	return sin((6.283185307179586 * z0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin((6.283185307179586d0 * z0))
end function

Java

public static double code(double z0) {
	return Math.sin((6.283185307179586 * z0));
}

Python

def code(z0):
	return math.sin((6.283185307179586 * z0))

Julia

function code(z0)
	return sin(Float64(6.283185307179586 * z0))
end

MATLAB

function tmp = code(z0)
	tmp = sin((6.283185307179586 * z0));
end

Wolfram

code[z0_] := N[Sin[N[(6.283185307179586 * z0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. pow1/3 N/A

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

   9. lift-PI.f64 N/A

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

   10. pow1/3 N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lower-cbrt.f64 52.9%

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

3. Applied rewrites 52.9%

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

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{1.4645918875615234} \cdot \left(z0 + z0\right)\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(1.4645918875615234 \cdot \left(z0 + z0\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot \left(z0 + z0\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \left(z0 + z0\right)\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(z0 + z0\right)}\right) \]

   5. count-2 N/A

      \[\leadsto \sin \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot \color{blue}{\left(2 \cdot z0\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{824491934883991}{562949953421312}\right) \cdot 2\right) \cdot z0\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sin \left(\left(\color{blue}{\frac{3982441812995697399929051632117}{1267650600228229401496703205376}} \cdot 2\right) \cdot z0\right) \]

   9. metadata-eval 53.4%

      \[\leadsto \sin \left(\color{blue}{6.283185307179586} \cdot z0\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(6.283185307179586 \cdot z0\right)} \]
8. Add Preprocessing

Alternative 10: 51.1% accurate, 2.9× speedup

Math

\[z0 \cdot \left(\left(\pi + \pi\right) - 1.3333333333333333 \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 z0
 (- (+ PI PI) (* 1.3333333333333333 (* (* (* z0 z0) PI) (* PI PI))))))

C

double code(double z0) {
	return z0 * ((((double) M_PI) + ((double) M_PI)) - (1.3333333333333333 * (((z0 * z0) * ((double) M_PI)) * (((double) M_PI) * ((double) M_PI)))));
}

Java

public static double code(double z0) {
	return z0 * ((Math.PI + Math.PI) - (1.3333333333333333 * (((z0 * z0) * Math.PI) * (Math.PI * Math.PI))));
}

Python

def code(z0):
	return z0 * ((math.pi + math.pi) - (1.3333333333333333 * (((z0 * z0) * math.pi) * (math.pi * math.pi))))

Julia

function code(z0)
	return Float64(z0 * Float64(Float64(pi + pi) - Float64(1.3333333333333333 * Float64(Float64(Float64(z0 * z0) * pi) * Float64(pi * pi)))))
end

MATLAB

function tmp = code(z0)
	tmp = z0 * ((pi + pi) - (1.3333333333333333 * (((z0 * z0) * pi) * (pi * pi))));
end

Wolfram

code[z0_] := N[(z0 * N[(N[(Pi + Pi), $MachinePrecision] - N[(1.3333333333333333 * N[(N[(N[(z0 * z0), $MachinePrecision] * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

2. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

      \[\leadsto z0 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \color{blue}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-PI.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   9. lower-PI.f64 51.1%

      \[\leadsto z0 \cdot \left(-1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right) \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{z0 \cdot \left(-1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + 2 \cdot \pi\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto z0 \cdot \left(\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right) + \color{blue}{2 \cdot \pi}\right) \]

   2. +-commutative N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi + \color{blue}{\frac{-4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi + \frac{-4}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   4. fp-cancel-sign-sub-inv N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   5. lower--.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto z0 \cdot \left(2 \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   7. count-2-rev N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \left(\mathsf{neg}\left(\frac{-4}{3}\right)\right) \cdot \color{blue}{\left({z0}^{2} \cdot {\pi}^{3}\right)}\right) \]

   10. metadata-eval 51.1%

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - 1.3333333333333333 \cdot \left(\color{blue}{{z0}^{2}} \cdot {\pi}^{3}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\pi}^{3}}\right)\right) \]

   12. lift-pow.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{\color{blue}{3}}\right)\right) \]

   13. cube-mult N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left({z0}^{2} \cdot \left(\pi \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left({z0}^{2} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(z0 \cdot \left(z0 \cdot \pi\right)\right) \cdot \left(\color{blue}{\pi} \cdot \pi\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(z0 \cdot \left(\pi \cdot z0\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(z0 \cdot \left(\pi \cdot z0\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]

   20. lower-*.f64 N/A

       \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \frac{4}{3} \cdot \left(\left(z0 \cdot \left(\pi \cdot z0\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right) \]

6. Applied rewrites 51.1%

   \[\leadsto z0 \cdot \left(\left(\pi + \pi\right) - \color{blue}{1.3333333333333333 \cdot \left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)}\right) \]
7. Add Preprocessing

Alternative 11: 51.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* 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. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 53.4%

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

3. Applied rewrites 53.4%

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

4. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.1%

      \[\leadsto 2 \cdot \left(z0 \cdot \pi\right) \]

6. Applied rewrites 51.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-*.f64 51.1%

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

8. Applied rewrites 51.1%

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

Reproduce

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