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

Percentage Accurate: 42.7% → 78.8%

Time: 2.9s

Alternatives: 11

Speedup: 11.8×

Specification

Math

\[\begin{array}{l} t_0 := \left(z1 \cdot \pi\right) \cdot z0\\ \frac{\sin t\_0}{\left(t\_0 \cdot z0\right) \cdot \pi} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* z1 PI) z0))) (/ (sin t_0) (* (* t_0 z0) PI))))

C

double code(double z1, double z0) {
	double t_0 = (z1 * ((double) M_PI)) * z0;
	return sin(t_0) / ((t_0 * z0) * ((double) M_PI));
}

Java

public static double code(double z1, double z0) {
	double t_0 = (z1 * Math.PI) * z0;
	return Math.sin(t_0) / ((t_0 * z0) * Math.PI);
}

Python

def code(z1, z0):
	t_0 = (z1 * math.pi) * z0
	return math.sin(t_0) / ((t_0 * z0) * math.pi)

Julia

function code(z1, z0)
	t_0 = Float64(Float64(z1 * pi) * z0)
	return Float64(sin(t_0) / Float64(Float64(t_0 * z0) * pi))
end

MATLAB

function tmp = code(z1, z0)
	t_0 = (z1 * pi) * z0;
	tmp = sin(t_0) / ((t_0 * z0) * pi);
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(z1 * Pi), $MachinePrecision] * z0), $MachinePrecision]}, N[(N[Sin[t$95$0], $MachinePrecision] / N[(N[(t$95$0 * z0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Initial Program: 42.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(z1 \cdot \pi\right) \cdot z0\\ \frac{\sin t\_0}{\left(t\_0 \cdot z0\right) \cdot \pi} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* z1 PI) z0))) (/ (sin t_0) (* (* t_0 z0) PI))))

C

double code(double z1, double z0) {
	double t_0 = (z1 * ((double) M_PI)) * z0;
	return sin(t_0) / ((t_0 * z0) * ((double) M_PI));
}

Java

public static double code(double z1, double z0) {
	double t_0 = (z1 * Math.PI) * z0;
	return Math.sin(t_0) / ((t_0 * z0) * Math.PI);
}

Python

def code(z1, z0):
	t_0 = (z1 * math.pi) * z0
	return math.sin(t_0) / ((t_0 * z0) * math.pi)

Julia

function code(z1, z0)
	t_0 = Float64(Float64(z1 * pi) * z0)
	return Float64(sin(t_0) / Float64(Float64(t_0 * z0) * pi))
end

MATLAB

function tmp = code(z1, z0)
	t_0 = (z1 * pi) * z0;
	tmp = sin(t_0) / ((t_0 * z0) * pi);
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(z1 * Pi), $MachinePrecision] * z0), $MachinePrecision]}, N[(N[Sin[t$95$0], $MachinePrecision] / N[(N[(t$95$0 * z0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \left(\pi \cdot \left|z1\right|\right)\\ t_1 := \left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := \left(t\_1 \cdot \left|z0\right|\right) \cdot \pi\\ t_3 := \frac{\sin t\_1}{t\_2}\\ t_4 := \left|z0\right| \cdot \left|z1\right|\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{-174}:\\ \;\;\;\;\frac{\frac{\sin \left(0.5 \cdot \pi - \left(\pi \cdot \left(t\_4 + 0.5\right) + \pi\right)\right) + \sin \left(t\_4 \cdot \pi\right)}{2}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\frac{\sin t\_0}{\left|z0\right| \cdot \pi}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \pi}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) (* PI (fabs z1))))
       (t_1 (* (* (fabs z1) PI) (fabs z0)))
       (t_2 (* (* t_1 (fabs z0)) PI))
       (t_3 (/ (sin t_1) t_2))
       (t_4 (* (fabs z0) (fabs z1))))
  (*
   (copysign 1.0 z0)
   (if (<= t_3 1e-174)
     (/
      (/
       (+
        (sin (- (* 0.5 PI) (+ (* PI (+ t_4 0.5)) PI)))
        (sin (* t_4 PI)))
       2.0)
      t_2)
     (if (<= t_3 INFINITY)
       (/ (/ (sin t_0) (* (fabs z0) PI)) t_0)
       (/ (sin PI) t_2))))))

C

double code(double z1, double z0) {
	double t_0 = fabs(z0) * (((double) M_PI) * fabs(z1));
	double t_1 = (fabs(z1) * ((double) M_PI)) * fabs(z0);
	double t_2 = (t_1 * fabs(z0)) * ((double) M_PI);
	double t_3 = sin(t_1) / t_2;
	double t_4 = fabs(z0) * fabs(z1);
	double tmp;
	if (t_3 <= 1e-174) {
		tmp = ((sin(((0.5 * ((double) M_PI)) - ((((double) M_PI) * (t_4 + 0.5)) + ((double) M_PI)))) + sin((t_4 * ((double) M_PI)))) / 2.0) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sin(t_0) / (fabs(z0) * ((double) M_PI))) / t_0;
	} else {
		tmp = sin(((double) M_PI)) / t_2;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.abs(z0) * (Math.PI * Math.abs(z1));
	double t_1 = (Math.abs(z1) * Math.PI) * Math.abs(z0);
	double t_2 = (t_1 * Math.abs(z0)) * Math.PI;
	double t_3 = Math.sin(t_1) / t_2;
	double t_4 = Math.abs(z0) * Math.abs(z1);
	double tmp;
	if (t_3 <= 1e-174) {
		tmp = ((Math.sin(((0.5 * Math.PI) - ((Math.PI * (t_4 + 0.5)) + Math.PI))) + Math.sin((t_4 * Math.PI))) / 2.0) / t_2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sin(t_0) / (Math.abs(z0) * Math.PI)) / t_0;
	} else {
		tmp = Math.sin(Math.PI) / t_2;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = math.fabs(z0) * (math.pi * math.fabs(z1))
	t_1 = (math.fabs(z1) * math.pi) * math.fabs(z0)
	t_2 = (t_1 * math.fabs(z0)) * math.pi
	t_3 = math.sin(t_1) / t_2
	t_4 = math.fabs(z0) * math.fabs(z1)
	tmp = 0
	if t_3 <= 1e-174:
		tmp = ((math.sin(((0.5 * math.pi) - ((math.pi * (t_4 + 0.5)) + math.pi))) + math.sin((t_4 * math.pi))) / 2.0) / t_2
	elif t_3 <= math.inf:
		tmp = (math.sin(t_0) / (math.fabs(z0) * math.pi)) / t_0
	else:
		tmp = math.sin(math.pi) / t_2
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(abs(z0) * Float64(pi * abs(z1)))
	t_1 = Float64(Float64(abs(z1) * pi) * abs(z0))
	t_2 = Float64(Float64(t_1 * abs(z0)) * pi)
	t_3 = Float64(sin(t_1) / t_2)
	t_4 = Float64(abs(z0) * abs(z1))
	tmp = 0.0
	if (t_3 <= 1e-174)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.5 * pi) - Float64(Float64(pi * Float64(t_4 + 0.5)) + pi))) + sin(Float64(t_4 * pi))) / 2.0) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sin(t_0) / Float64(abs(z0) * pi)) / t_0);
	else
		tmp = Float64(sin(pi) / t_2);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = abs(z0) * (pi * abs(z1));
	t_1 = (abs(z1) * pi) * abs(z0);
	t_2 = (t_1 * abs(z0)) * pi;
	t_3 = sin(t_1) / t_2;
	t_4 = abs(z0) * abs(z1);
	tmp = 0.0;
	if (t_3 <= 1e-174)
		tmp = ((sin(((0.5 * pi) - ((pi * (t_4 + 0.5)) + pi))) + sin((t_4 * pi))) / 2.0) / t_2;
	elseif (t_3 <= Inf)
		tmp = (sin(t_0) / (abs(z0) * pi)) / t_0;
	else
		tmp = sin(pi) / t_2;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * N[(Pi * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t$95$1], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 1e-174], N[(N[(N[(N[Sin[N[(N[(0.5 * Pi), $MachinePrecision] - N[(N[(Pi * N[(t$95$4 + 0.5), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(t$95$4 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sin[t$95$0], $MachinePrecision] / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sin[Pi], $MachinePrecision] / t$95$2), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 (*.f64 (*.f64 z1 (PI.f64)) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z1 (PI.f64)) z0) z0) (PI.f64))) < 1e-174

   1. Initial program 42.7%

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

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot 1}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)} \cdot 1}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(z0 \cdot \left(\pi \cdot z1\right) - \pi \cdot 0.5\right) - \left(-\sin \left(z0 \cdot \left(\pi \cdot z1\right)\right)\right)}{2}}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   5. Applied rewrites 37.8%

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(0.5 \cdot \pi - \left(\pi \cdot \left(z0 \cdot z1 + 0.5\right) + \pi\right)\right) + \sin \left(\left(z0 \cdot z1\right) \cdot \pi\right)}{2}}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   if 1e-174 < (/.f64 (sin.f64 (*.f64 (*.f64 z1 (PI.f64)) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z1 (PI.f64)) z0) z0) (PI.f64))) < +inf.0

   1. Initial program 42.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 53.7%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 53.7%

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

   3. Applied rewrites 53.7%

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

   if +inf.0 < (/.f64 (sin.f64 (*.f64 (*.f64 z1 (PI.f64)) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z1 (PI.f64)) z0) z0) (PI.f64)))

   1. Initial program 42.7%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. cos-+PI/2-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

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

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   3. Applied rewrites 18.5%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(-z1\right) \cdot \left(z0 \cdot \pi\right) + \pi \cdot 0.5\right) + \pi \cdot 0.5\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-PI.f64 29.5%

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

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\sin \color{blue}{\pi}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.8% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log \left(e^{\pi \cdot \left|z0\right|}\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 8.6e-7)
   (/ 1.0 (* (fabs z0) PI))
   (/ 1.0 (log (exp (* PI (fabs z0))))))))

C

double code(double z1, double z0) {
	double tmp;
	if (fabs(z0) <= 8.6e-7) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else {
		tmp = 1.0 / log(exp((((double) M_PI) * fabs(z0))));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.abs(z0) <= 8.6e-7) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else {
		tmp = 1.0 / Math.log(Math.exp((Math.PI * Math.abs(z0))));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.fabs(z0) <= 8.6e-7:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	else:
		tmp = 1.0 / math.log(math.exp((math.pi * math.fabs(z0))))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (abs(z0) <= 8.6e-7)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	else
		tmp = Float64(1.0 / log(exp(Float64(pi * abs(z0)))));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (abs(z0) <= 8.6e-7)
		tmp = 1.0 / (abs(z0) * pi);
	else
		tmp = 1.0 / log(exp((pi * abs(z0))));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 8.6e-7], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Log[N[Exp[N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 8.6000000000000002e-7

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 8.6000000000000002e-7 < z0

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \color{blue}{\pi}} \]

      2. lift-PI.f64 N/A

         \[\leadsto \frac{1}{z0 \cdot \mathsf{PI}\left(\right)} \]

      3. add-log-exp N/A

         \[\leadsto \frac{1}{z0 \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

      4. log-pow-rev N/A

         \[\leadsto \frac{1}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{z0}\right)} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{1}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{z0}\right)} \]

      6. lift-PI.f64 N/A

         \[\leadsto \frac{1}{\log \left({\left(e^{\pi}\right)}^{z0}\right)} \]

      7. pow-exp N/A

         \[\leadsto \frac{1}{\log \left(e^{\pi \cdot z0}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\log \left(e^{z0 \cdot \pi}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\log \left(e^{z0 \cdot \pi}\right)} \]

      10. lower-exp.f64 38.3%

          \[\leadsto \frac{1}{\log \left(e^{z0 \cdot \pi}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\log \left(e^{z0 \cdot \pi}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\log \left(e^{\pi \cdot z0}\right)} \]

      13. lower-*.f64 38.3%

          \[\leadsto \frac{1}{\log \left(e^{\pi \cdot z0}\right)} \]

   6. Applied rewrites 38.3%

      \[\leadsto \frac{1}{\log \left(e^{\pi \cdot z0}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \left(\pi \cdot \left|z1\right|\right)\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 3.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{elif}\;\left|z0\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sin t\_0}{\pi} \cdot \frac{1}{t\_0 \cdot \left|z0\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}{\left(\left(\left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) (* PI (fabs z1)))))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 3.7e-53)
     (/ 1.0 (* (fabs z0) PI))
     (if (<= (fabs z0) 3.5e+152)
       (* (/ (sin t_0) PI) (/ 1.0 (* t_0 (fabs z0))))
       (/
        (* 0.5 (cos (- (* 0.5 PI))))
        (* (* (* (* (fabs z1) PI) (fabs z0)) (fabs z0)) PI)))))))

C

double code(double z1, double z0) {
	double t_0 = fabs(z0) * (((double) M_PI) * fabs(z1));
	double tmp;
	if (fabs(z0) <= 3.7e-53) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else if (fabs(z0) <= 3.5e+152) {
		tmp = (sin(t_0) / ((double) M_PI)) * (1.0 / (t_0 * fabs(z0)));
	} else {
		tmp = (0.5 * cos(-(0.5 * ((double) M_PI)))) / ((((fabs(z1) * ((double) M_PI)) * fabs(z0)) * fabs(z0)) * ((double) M_PI));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.abs(z0) * (Math.PI * Math.abs(z1));
	double tmp;
	if (Math.abs(z0) <= 3.7e-53) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else if (Math.abs(z0) <= 3.5e+152) {
		tmp = (Math.sin(t_0) / Math.PI) * (1.0 / (t_0 * Math.abs(z0)));
	} else {
		tmp = (0.5 * Math.cos(-(0.5 * Math.PI))) / ((((Math.abs(z1) * Math.PI) * Math.abs(z0)) * Math.abs(z0)) * Math.PI);
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = math.fabs(z0) * (math.pi * math.fabs(z1))
	tmp = 0
	if math.fabs(z0) <= 3.7e-53:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	elif math.fabs(z0) <= 3.5e+152:
		tmp = (math.sin(t_0) / math.pi) * (1.0 / (t_0 * math.fabs(z0)))
	else:
		tmp = (0.5 * math.cos(-(0.5 * math.pi))) / ((((math.fabs(z1) * math.pi) * math.fabs(z0)) * math.fabs(z0)) * math.pi)
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(abs(z0) * Float64(pi * abs(z1)))
	tmp = 0.0
	if (abs(z0) <= 3.7e-53)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	elseif (abs(z0) <= 3.5e+152)
		tmp = Float64(Float64(sin(t_0) / pi) * Float64(1.0 / Float64(t_0 * abs(z0))));
	else
		tmp = Float64(Float64(0.5 * cos(Float64(-Float64(0.5 * pi)))) / Float64(Float64(Float64(Float64(abs(z1) * pi) * abs(z0)) * abs(z0)) * pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = abs(z0) * (pi * abs(z1));
	tmp = 0.0;
	if (abs(z0) <= 3.7e-53)
		tmp = 1.0 / (abs(z0) * pi);
	elseif (abs(z0) <= 3.5e+152)
		tmp = (sin(t_0) / pi) * (1.0 / (t_0 * abs(z0)));
	else
		tmp = (0.5 * cos(-(0.5 * pi))) / ((((abs(z1) * pi) * abs(z0)) * abs(z0)) * pi);
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * N[(Pi * N[Abs[z1], $MachinePrecision]), $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], 3.7e-53], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 3.5e+152], N[(N[(N[Sin[t$95$0], $MachinePrecision] / Pi), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[(-N[(0.5 * Pi), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z0 < 3.6999999999999998e-53

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 3.6999999999999998e-53 < z0 < 3.4999999999999998e152

   1. Initial program 42.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. mult-flip N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)}{\pi}} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(z1 \cdot \pi\right) \cdot z0\right)}}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin \color{blue}{\left(z0 \cdot \left(z1 \cdot \pi\right)\right)}}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin \color{blue}{\left(z0 \cdot \left(z1 \cdot \pi\right)\right)}}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sin \left(z0 \cdot \color{blue}{\left(z1 \cdot \pi\right)}\right)}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin \left(z0 \cdot \color{blue}{\left(\pi \cdot z1\right)}\right)}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(z0 \cdot \color{blue}{\left(\pi \cdot z1\right)}\right)}{\pi} \cdot \frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0} \]

      14. lower-/.f64 41.7%

          \[\leadsto \frac{\sin \left(z0 \cdot \left(\pi \cdot z1\right)\right)}{\pi} \cdot \color{blue}{\frac{1}{\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0}} \]

   3. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\sin \left(z0 \cdot \left(\pi \cdot z1\right)\right)}{\pi} \cdot \frac{1}{\left(z0 \cdot \left(\pi \cdot z1\right)\right) \cdot z0}} \]

   if 3.4999999999999998e152 < z0

   1. Initial program 42.7%

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

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot 1}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)} \cdot 1}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(z0 \cdot \left(\pi \cdot z1\right) - \pi \cdot 0.5\right) - \left(-\sin \left(z0 \cdot \left(\pi \cdot z1\right)\right)\right)}{2}}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      2. lower-cos.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 29.5%

         \[\leadsto \frac{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\\ t_1 := \left(t\_0 \cdot \left|z0\right|\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{elif}\;\left|z0\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sin t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* (fabs z1) PI) (fabs z0)))
       (t_1 (* (* t_0 (fabs z0)) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.6e-53)
     (/ 1.0 (* (fabs z0) PI))
     (if (<= (fabs z0) 3.5e+152)
       (/ (sin t_0) t_1)
       (/ (* 0.5 (cos (- (* 0.5 PI)))) t_1))))))

C

double code(double z1, double z0) {
	double t_0 = (fabs(z1) * ((double) M_PI)) * fabs(z0);
	double t_1 = (t_0 * fabs(z0)) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.6e-53) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else if (fabs(z0) <= 3.5e+152) {
		tmp = sin(t_0) / t_1;
	} else {
		tmp = (0.5 * cos(-(0.5 * ((double) M_PI)))) / t_1;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (Math.abs(z1) * Math.PI) * Math.abs(z0);
	double t_1 = (t_0 * Math.abs(z0)) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.6e-53) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else if (Math.abs(z0) <= 3.5e+152) {
		tmp = Math.sin(t_0) / t_1;
	} else {
		tmp = (0.5 * Math.cos(-(0.5 * Math.PI))) / t_1;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = (math.fabs(z1) * math.pi) * math.fabs(z0)
	t_1 = (t_0 * math.fabs(z0)) * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.6e-53:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	elif math.fabs(z0) <= 3.5e+152:
		tmp = math.sin(t_0) / t_1
	else:
		tmp = (0.5 * math.cos(-(0.5 * math.pi))) / t_1
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(abs(z1) * pi) * abs(z0))
	t_1 = Float64(Float64(t_0 * abs(z0)) * pi)
	tmp = 0.0
	if (abs(z0) <= 1.6e-53)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	elseif (abs(z0) <= 3.5e+152)
		tmp = Float64(sin(t_0) / t_1);
	else
		tmp = Float64(Float64(0.5 * cos(Float64(-Float64(0.5 * pi)))) / t_1);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (abs(z1) * pi) * abs(z0);
	t_1 = (t_0 * abs(z0)) * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.6e-53)
		tmp = 1.0 / (abs(z0) * pi);
	elseif (abs(z0) <= 3.5e+152)
		tmp = sin(t_0) / t_1;
	else
		tmp = (0.5 * cos(-(0.5 * pi))) / t_1;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $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], 1.6e-53], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 3.5e+152], N[(N[Sin[t$95$0], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * N[Cos[(-N[(0.5 * Pi), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 1.6e-53

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 1.6e-53 < z0 < 3.4999999999999998e152

   1. Initial program 42.7%

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

   if 3.4999999999999998e152 < z0

   1. Initial program 42.7%

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

      1. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot 1}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right)} \cdot 1}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      3. sin-PI/2 N/A

         \[\leadsto \frac{\sin \left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      4. sin-mult N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(z0 \cdot \left(\pi \cdot z1\right) - \pi \cdot 0.5\right) - \left(-\sin \left(z0 \cdot \left(\pi \cdot z1\right)\right)\right)}{2}}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      2. lower-cos.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 29.5%

         \[\leadsto \frac{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \cos \left(-0.5 \cdot \pi\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\\ t_1 := \left(t\_0 \cdot \left|z0\right|\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{elif}\;\left|z0\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sin t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \pi}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* (fabs z1) PI) (fabs z0)))
       (t_1 (* (* t_0 (fabs z0)) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.6e-53)
     (/ 1.0 (* (fabs z0) PI))
     (if (<= (fabs z0) 3.5e+152)
       (/ (sin t_0) t_1)
       (/ (sin PI) t_1))))))

C

double code(double z1, double z0) {
	double t_0 = (fabs(z1) * ((double) M_PI)) * fabs(z0);
	double t_1 = (t_0 * fabs(z0)) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.6e-53) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else if (fabs(z0) <= 3.5e+152) {
		tmp = sin(t_0) / t_1;
	} else {
		tmp = sin(((double) M_PI)) / t_1;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (Math.abs(z1) * Math.PI) * Math.abs(z0);
	double t_1 = (t_0 * Math.abs(z0)) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.6e-53) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else if (Math.abs(z0) <= 3.5e+152) {
		tmp = Math.sin(t_0) / t_1;
	} else {
		tmp = Math.sin(Math.PI) / t_1;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = (math.fabs(z1) * math.pi) * math.fabs(z0)
	t_1 = (t_0 * math.fabs(z0)) * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.6e-53:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	elif math.fabs(z0) <= 3.5e+152:
		tmp = math.sin(t_0) / t_1
	else:
		tmp = math.sin(math.pi) / t_1
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(abs(z1) * pi) * abs(z0))
	t_1 = Float64(Float64(t_0 * abs(z0)) * pi)
	tmp = 0.0
	if (abs(z0) <= 1.6e-53)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	elseif (abs(z0) <= 3.5e+152)
		tmp = Float64(sin(t_0) / t_1);
	else
		tmp = Float64(sin(pi) / t_1);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (abs(z1) * pi) * abs(z0);
	t_1 = (t_0 * abs(z0)) * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.6e-53)
		tmp = 1.0 / (abs(z0) * pi);
	elseif (abs(z0) <= 3.5e+152)
		tmp = sin(t_0) / t_1;
	else
		tmp = sin(pi) / t_1;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $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], 1.6e-53], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 3.5e+152], N[(N[Sin[t$95$0], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sin[Pi], $MachinePrecision] / t$95$1), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 1.6e-53

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 1.6e-53 < z0 < 3.4999999999999998e152

   1. Initial program 42.7%

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

   if 3.4999999999999998e152 < z0

   1. Initial program 42.7%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. cos-+PI/2-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

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

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   3. Applied rewrites 18.5%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(-z1\right) \cdot \left(z0 \cdot \pi\right) + \pi \cdot 0.5\right) + \pi \cdot 0.5\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-PI.f64 29.5%

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

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\sin \color{blue}{\pi}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\\ t_1 := \left|z0\right| \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{t\_1}\\ \mathbf{elif}\;\left|z0\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sin t\_0}{\left(\left|z0\right| \cdot \left(\pi \cdot \left|z1\right|\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \pi}{\left(t\_0 \cdot \left|z0\right|\right) \cdot \pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* (fabs z1) PI) (fabs z0))) (t_1 (* (fabs z0) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.6e-53)
     (/ 1.0 t_1)
     (if (<= (fabs z0) 3.5e+152)
       (/ (sin t_0) (* (* (fabs z0) (* PI (fabs z1))) t_1))
       (/ (sin PI) (* (* t_0 (fabs z0)) PI)))))))

C

double code(double z1, double z0) {
	double t_0 = (fabs(z1) * ((double) M_PI)) * fabs(z0);
	double t_1 = fabs(z0) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.6e-53) {
		tmp = 1.0 / t_1;
	} else if (fabs(z0) <= 3.5e+152) {
		tmp = sin(t_0) / ((fabs(z0) * (((double) M_PI) * fabs(z1))) * t_1);
	} else {
		tmp = sin(((double) M_PI)) / ((t_0 * fabs(z0)) * ((double) M_PI));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (Math.abs(z1) * Math.PI) * Math.abs(z0);
	double t_1 = Math.abs(z0) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.6e-53) {
		tmp = 1.0 / t_1;
	} else if (Math.abs(z0) <= 3.5e+152) {
		tmp = Math.sin(t_0) / ((Math.abs(z0) * (Math.PI * Math.abs(z1))) * t_1);
	} else {
		tmp = Math.sin(Math.PI) / ((t_0 * Math.abs(z0)) * Math.PI);
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = (math.fabs(z1) * math.pi) * math.fabs(z0)
	t_1 = math.fabs(z0) * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.6e-53:
		tmp = 1.0 / t_1
	elif math.fabs(z0) <= 3.5e+152:
		tmp = math.sin(t_0) / ((math.fabs(z0) * (math.pi * math.fabs(z1))) * t_1)
	else:
		tmp = math.sin(math.pi) / ((t_0 * math.fabs(z0)) * math.pi)
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(abs(z1) * pi) * abs(z0))
	t_1 = Float64(abs(z0) * pi)
	tmp = 0.0
	if (abs(z0) <= 1.6e-53)
		tmp = Float64(1.0 / t_1);
	elseif (abs(z0) <= 3.5e+152)
		tmp = Float64(sin(t_0) / Float64(Float64(abs(z0) * Float64(pi * abs(z1))) * t_1));
	else
		tmp = Float64(sin(pi) / Float64(Float64(t_0 * abs(z0)) * pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (abs(z1) * pi) * abs(z0);
	t_1 = abs(z0) * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.6e-53)
		tmp = 1.0 / t_1;
	elseif (abs(z0) <= 3.5e+152)
		tmp = sin(t_0) / ((abs(z0) * (pi * abs(z1))) * t_1);
	else
		tmp = sin(pi) / ((t_0 * abs(z0)) * pi);
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $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], 1.6e-53], N[(1.0 / t$95$1), $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 3.5e+152], N[(N[Sin[t$95$0], $MachinePrecision] / N[(N[(N[Abs[z0], $MachinePrecision] * N[(Pi * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[Pi], $MachinePrecision] / N[(N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < 1.6e-53

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 1.6e-53 < z0 < 3.4999999999999998e152

   1. Initial program 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 42.7%

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

   3. Applied rewrites 42.7%

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

   if 3.4999999999999998e152 < z0

   1. Initial program 42.7%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. cos-+PI/2-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

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

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   3. Applied rewrites 18.5%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(-z1\right) \cdot \left(z0 \cdot \pi\right) + \pi \cdot 0.5\right) + \pi \cdot 0.5\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-PI.f64 29.5%

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

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\sin \color{blue}{\pi}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|z0\right| \cdot \left|z1\right|\right) \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{elif}\;\left|z0\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sin t\_0}{\left(t\_0 \cdot \left|z0\right|\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \pi}{\left(\left(\left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (* (fabs z0) (fabs z1)) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 1.6e-53)
     (/ 1.0 (* (fabs z0) PI))
     (if (<= (fabs z0) 3.5e+152)
       (/ (sin t_0) (* (* t_0 (fabs z0)) PI))
       (/
        (sin PI)
        (* (* (* (* (fabs z1) PI) (fabs z0)) (fabs z0)) PI)))))))

C

double code(double z1, double z0) {
	double t_0 = (fabs(z0) * fabs(z1)) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 1.6e-53) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else if (fabs(z0) <= 3.5e+152) {
		tmp = sin(t_0) / ((t_0 * fabs(z0)) * ((double) M_PI));
	} else {
		tmp = sin(((double) M_PI)) / ((((fabs(z1) * ((double) M_PI)) * fabs(z0)) * fabs(z0)) * ((double) M_PI));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = (Math.abs(z0) * Math.abs(z1)) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 1.6e-53) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else if (Math.abs(z0) <= 3.5e+152) {
		tmp = Math.sin(t_0) / ((t_0 * Math.abs(z0)) * Math.PI);
	} else {
		tmp = Math.sin(Math.PI) / ((((Math.abs(z1) * Math.PI) * Math.abs(z0)) * Math.abs(z0)) * Math.PI);
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	t_0 = (math.fabs(z0) * math.fabs(z1)) * math.pi
	tmp = 0
	if math.fabs(z0) <= 1.6e-53:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	elif math.fabs(z0) <= 3.5e+152:
		tmp = math.sin(t_0) / ((t_0 * math.fabs(z0)) * math.pi)
	else:
		tmp = math.sin(math.pi) / ((((math.fabs(z1) * math.pi) * math.fabs(z0)) * math.fabs(z0)) * math.pi)
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(abs(z0) * abs(z1)) * pi)
	tmp = 0.0
	if (abs(z0) <= 1.6e-53)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	elseif (abs(z0) <= 3.5e+152)
		tmp = Float64(sin(t_0) / Float64(Float64(t_0 * abs(z0)) * pi));
	else
		tmp = Float64(sin(pi) / Float64(Float64(Float64(Float64(abs(z1) * pi) * abs(z0)) * abs(z0)) * pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = (abs(z0) * abs(z1)) * pi;
	tmp = 0.0;
	if (abs(z0) <= 1.6e-53)
		tmp = 1.0 / (abs(z0) * pi);
	elseif (abs(z0) <= 3.5e+152)
		tmp = sin(t_0) / ((t_0 * abs(z0)) * pi);
	else
		tmp = sin(pi) / ((((abs(z1) * pi) * abs(z0)) * abs(z0)) * pi);
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $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], 1.6e-53], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z0], $MachinePrecision], 3.5e+152], N[(N[Sin[t$95$0], $MachinePrecision] / N[(N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[Sin[Pi], $MachinePrecision] / N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z0 < 1.6e-53

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 1.6e-53 < z0 < 3.4999999999999998e152

   1. Initial program 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 42.7%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.7%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 42.7%

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

   3. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 42.6%

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

   5. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 42.7%

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

   7. Applied rewrites 42.7%

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

   if 3.4999999999999998e152 < z0

   1. Initial program 42.7%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. cos-+PI/2-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

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

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   3. Applied rewrites 18.5%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(-z1\right) \cdot \left(z0 \cdot \pi\right) + \pi \cdot 0.5\right) + \pi \cdot 0.5\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-PI.f64 29.5%

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

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\sin \color{blue}{\pi}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.4% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 3 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\left|z0\right| \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \pi}{\left(\left(\left(\left|z1\right| \cdot \pi\right) \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi}\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 3e+63)
   (/ 1.0 (* (fabs z0) PI))
   (/ (sin PI) (* (* (* (* (fabs z1) PI) (fabs z0)) (fabs z0)) PI)))))

C

double code(double z1, double z0) {
	double tmp;
	if (fabs(z0) <= 3e+63) {
		tmp = 1.0 / (fabs(z0) * ((double) M_PI));
	} else {
		tmp = sin(((double) M_PI)) / ((((fabs(z1) * ((double) M_PI)) * fabs(z0)) * fabs(z0)) * ((double) M_PI));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.abs(z0) <= 3e+63) {
		tmp = 1.0 / (Math.abs(z0) * Math.PI);
	} else {
		tmp = Math.sin(Math.PI) / ((((Math.abs(z1) * Math.PI) * Math.abs(z0)) * Math.abs(z0)) * Math.PI);
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.fabs(z0) <= 3e+63:
		tmp = 1.0 / (math.fabs(z0) * math.pi)
	else:
		tmp = math.sin(math.pi) / ((((math.fabs(z1) * math.pi) * math.fabs(z0)) * math.fabs(z0)) * math.pi)
	return math.copysign(1.0, z0) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (abs(z0) <= 3e+63)
		tmp = Float64(1.0 / Float64(abs(z0) * pi));
	else
		tmp = Float64(sin(pi) / Float64(Float64(Float64(Float64(abs(z1) * pi) * abs(z0)) * abs(z0)) * pi));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (abs(z0) <= 3e+63)
		tmp = 1.0 / (abs(z0) * pi);
	else
		tmp = sin(pi) / ((((abs(z1) * pi) * abs(z0)) * abs(z0)) * pi);
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 3e+63], N[(1.0 / N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[Sin[Pi], $MachinePrecision] / N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 3e63

   1. Initial program 42.7%

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

   2. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 51.7%

         \[\leadsto \frac{1}{z0 \cdot \pi} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   if 3e63 < z0

   1. Initial program 42.7%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. cos-+PI/2-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

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

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(z1 \cdot \pi\right) \cdot z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   3. Applied rewrites 18.5%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(\left(-z1\right) \cdot \left(z0 \cdot \pi\right) + \pi \cdot 0.5\right) + \pi \cdot 0.5\right)}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-PI.f64 29.5%

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

   6. Applied rewrites 29.5%

      \[\leadsto \frac{\sin \color{blue}{\pi}}{\left(\left(\left(z1 \cdot \pi\right) \cdot z0\right) \cdot z0\right) \cdot \pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.8% accurate, 6.5× speedup

Math

\[\frac{2}{\left(\pi \cdot z0\right) \cdot 2} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 2.0 (* (* PI z0) 2.0)))

C

double code(double z1, double z0) {
	return 2.0 / ((((double) M_PI) * z0) * 2.0);
}

Java

public static double code(double z1, double z0) {
	return 2.0 / ((Math.PI * z0) * 2.0);
}

Python

def code(z1, z0):
	return 2.0 / ((math.pi * z0) * 2.0)

Julia

function code(z1, z0)
	return Float64(2.0 / Float64(Float64(pi * z0) * 2.0))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 2.0 / ((pi * z0) * 2.0);
end

Wolfram

code[z1_, z0_] := N[(2.0 / N[(N[(Pi * z0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.7%

      \[\leadsto \frac{1}{z0 \cdot \pi} \]

4. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
5. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \color{blue}{1} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot 1 \]

   4. cosh-0-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \cosh 0 \]

   5. cosh-def N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{2}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{2 \cdot \color{blue}{1}} \]

   7. cosh-0-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{2 \cdot \cosh 0} \]

   8. cosh-undef-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + \color{blue}{e^{\mathsf{neg}\left(0\right)}}} \]

   9. frac-times N/A

      \[\leadsto \frac{1 \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}{\color{blue}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

   10. cosh-undef-rev N/A

       \[\leadsto \frac{1 \cdot \left(2 \cdot \cosh 0\right)}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   11. cosh-0-rev N/A

       \[\leadsto \frac{1 \cdot \left(2 \cdot 1\right)}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1 \cdot 2}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   15. cosh-0-rev N/A

       \[\leadsto \frac{2 \cdot \cosh 0}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   16. cosh-undef-rev N/A

       \[\leadsto \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

   18. cosh-undef-rev N/A

       \[\leadsto \frac{2 \cdot \cosh 0}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   19. cosh-0-rev N/A

       \[\leadsto \frac{2 \cdot 1}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   20. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(z0 \cdot \pi\right) \cdot \color{blue}{\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

6. Applied rewrites 51.8%

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

Alternative 10: 51.7% accurate, 8.4× speedup

Math

\[\frac{1}{z0 \cdot \pi} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 1.0 (* z0 PI)))

C

double code(double z1, double z0) {
	return 1.0 / (z0 * ((double) M_PI));
}

Java

public static double code(double z1, double z0) {
	return 1.0 / (z0 * Math.PI);
}

Python

def code(z1, z0):
	return 1.0 / (z0 * math.pi)

Julia

function code(z1, z0)
	return Float64(1.0 / Float64(z0 * pi))
end

MATLAB

function tmp = code(z1, z0)
	tmp = 1.0 / (z0 * pi);
end

Wolfram

code[z1_, z0_] := N[(1.0 / N[(z0 * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.7%

      \[\leadsto \frac{1}{z0 \cdot \pi} \]

4. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
5. Add Preprocessing

Alternative 11: 51.6% accurate, 11.8× speedup

Math

\[\frac{0.3183098861837907}{z0} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (/ 0.3183098861837907 z0))

C

double code(double z1, double z0) {
	return 0.3183098861837907 / 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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = 0.3183098861837907d0 / z0
end function

Java

public static double code(double z1, double z0) {
	return 0.3183098861837907 / z0;
}

Python

def code(z1, z0):
	return 0.3183098861837907 / z0

Julia

function code(z1, z0)
	return Float64(0.3183098861837907 / z0)
end

MATLAB

function tmp = code(z1, z0)
	tmp = 0.3183098861837907 / z0;
end

Wolfram

code[z1_, z0_] := N[(0.3183098861837907 / z0), $MachinePrecision]

Derivation

1. Initial program 42.7%

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

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.7%

      \[\leadsto \frac{1}{z0 \cdot \pi} \]

4. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{1}{z0 \cdot \pi}} \]
5. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{z0 \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \color{blue}{1} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot 1 \]

   4. cosh-0-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \cosh 0 \]

   5. cosh-def N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{2}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{2 \cdot \color{blue}{1}} \]

   7. cosh-0-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{2 \cdot \cosh 0} \]

   8. cosh-undef-rev N/A

      \[\leadsto \frac{1}{z0 \cdot \pi} \cdot \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + \color{blue}{e^{\mathsf{neg}\left(0\right)}}} \]

   9. frac-times N/A

      \[\leadsto \frac{1 \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}{\color{blue}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

   10. cosh-undef-rev N/A

       \[\leadsto \frac{1 \cdot \left(2 \cdot \cosh 0\right)}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   11. cosh-0-rev N/A

       \[\leadsto \frac{1 \cdot \left(2 \cdot 1\right)}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1 \cdot 2}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   15. cosh-0-rev N/A

       \[\leadsto \frac{2 \cdot \cosh 0}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   16. cosh-undef-rev N/A

       \[\leadsto \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   17. lower-/.f64 N/A

       \[\leadsto \frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\color{blue}{\left(z0 \cdot \pi\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

   18. cosh-undef-rev N/A

       \[\leadsto \frac{2 \cdot \cosh 0}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   19. cosh-0-rev N/A

       \[\leadsto \frac{2 \cdot 1}{\left(z0 \cdot \color{blue}{\pi}\right) \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   20. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(z0 \cdot \pi\right)} \cdot \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(z0 \cdot \pi\right) \cdot \color{blue}{\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}} \]

6. Applied rewrites 51.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{2}}{\color{blue}{\pi \cdot z0}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{\color{blue}{\pi} \cdot z0} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\pi \cdot \color{blue}{z0}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{z0}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{z0}} \]

   9. lower-/.f64 51.6%

      \[\leadsto \frac{\frac{1}{\pi}}{z0} \]

8. Applied rewrites 51.6%

   \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{z0}} \]

9. Evaluated real constant 51.6%

   \[\leadsto \frac{0.3183098861837907}{z0} \]
10. Add Preprocessing

Reproduce

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