Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 99.8%

Time: 11.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \frac{\sin y}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{-8}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z\_m}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1e-8) (/ x (* y (/ z_m (sin y)))) (/ (* x (/ (sin y) y)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1e-8) {
		tmp = x / (y * (z_m / sin(y)));
	} else {
		tmp = (x * (sin(y) / y)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1d-8) then
        tmp = x / (y * (z_m / sin(y)))
    else
        tmp = (x * (sin(y) / y)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1e-8) {
		tmp = x / (y * (z_m / Math.sin(y)));
	} else {
		tmp = (x * (Math.sin(y) / y)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1e-8:
		tmp = x / (y * (z_m / math.sin(y)))
	else:
		tmp = (x * (math.sin(y) / y)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1e-8)
		tmp = Float64(x / Float64(y * Float64(z_m / sin(y))));
	else
		tmp = Float64(Float64(x * Float64(sin(y) / y)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1e-8)
		tmp = x / (y * (z_m / sin(y)));
	else
		tmp = (x * (sin(y) / y)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1e-8], N[(x / N[(y * N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e-8

   1. Initial program 93.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}}}{z} \cdot x \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

      10. lower-*.f64 88.6

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z}} \cdot x \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot z}{\sin y}}} \cdot x \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot z}{\sin y}}{x}}} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot z}{\sin y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot z}{\sin y}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot z}}{\sin y}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{z}{\sin y}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{z}{\sin y}}} \]

      10. lower-/.f64 91.2

          \[\leadsto \frac{x}{y \cdot \color{blue}{\frac{z}{\sin y}}} \]

   6. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \frac{z}{\sin y}}} \]

   if 1e-8 < z

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999:\\ \;\;\;\;\frac{x}{y \cdot \frac{z\_m}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999)
    (/ x (* y (/ z_m (sin y))))
    (* (/ (fma y (* -0.16666666666666666 (* y y)) y) y) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999) {
		tmp = x / (y * (z_m / sin(y)));
	} else {
		tmp = (fma(y, (-0.16666666666666666 * (y * y)), y) / y) * (x / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999)
		tmp = Float64(x / Float64(y * Float64(z_m / sin(y))));
	else
		tmp = Float64(Float64(fma(y, Float64(-0.16666666666666666 * Float64(y * y)), y) / y) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999], N[(x / N[(y * N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.998999999999999999

   1. Initial program 90.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}}}{z} \cdot x \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

      10. lower-*.f64 92.7

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z}} \cdot x \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot z}{\sin y}}} \cdot x \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y \cdot z}{\sin y}}{x}}} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot z}{\sin y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot z}{\sin y}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot z}}{\sin y}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{z}{\sin y}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{z}{\sin y}}} \]

      10. lower-/.f64 92.8

          \[\leadsto \frac{x}{y \cdot \color{blue}{\frac{z}{\sin y}}} \]

   6. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \frac{z}{\sin y}}} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 71.4

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)}}{z} \cdot \frac{x}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right) + y \cdot 1}}{z} \cdot \frac{x}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \frac{-1}{6}} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{z} \cdot \frac{x}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y \cdot {y}^{2}, y\right)}}{z} \cdot \frac{x}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right)}{z} \cdot \frac{x}{y} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

      10. lower-*.f64 71.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

   7. Applied rewrites 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}}{z} \cdot \frac{x}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \frac{x}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z}} \cdot \frac{x}{y} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \color{blue}{\frac{x}{y}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{z \cdot y}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{\color{blue}{y \cdot z}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \color{blue}{\frac{x}{z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      9. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}} \cdot \frac{x}{z} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999:\\ \;\;\;\;x \cdot \frac{\sin y}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999)
    (* x (/ (sin y) (* z_m y)))
    (* (/ (fma y (* -0.16666666666666666 (* y y)) y) y) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999) {
		tmp = x * (sin(y) / (z_m * y));
	} else {
		tmp = (fma(y, (-0.16666666666666666 * (y * y)), y) / y) * (x / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999)
		tmp = Float64(x * Float64(sin(y) / Float64(z_m * y)));
	else
		tmp = Float64(Float64(fma(y, Float64(-0.16666666666666666 * Float64(y * y)), y) / y) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999], N[(x * N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.998999999999999999

   1. Initial program 90.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}}}{z} \cdot x \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z \cdot y}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

      10. lower-*.f64 92.7

          \[\leadsto \frac{\sin y}{\color{blue}{y \cdot z}} \cdot x \]

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\frac{\sin y}{y \cdot z} \cdot x} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 71.4

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)}}{z} \cdot \frac{x}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right) + y \cdot 1}}{z} \cdot \frac{x}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \frac{-1}{6}} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{z} \cdot \frac{x}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y \cdot {y}^{2}, y\right)}}{z} \cdot \frac{x}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right)}{z} \cdot \frac{x}{y} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

      10. lower-*.f64 71.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

   7. Applied rewrites 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}}{z} \cdot \frac{x}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \frac{x}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z}} \cdot \frac{x}{y} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \color{blue}{\frac{x}{y}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{z \cdot y}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{\color{blue}{y \cdot z}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \color{blue}{\frac{x}{z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      9. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}} \cdot \frac{x}{z} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999:\\ \;\;\;\;\sin y \cdot \frac{x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999)
    (* (sin y) (/ x (* z_m y)))
    (* (/ (fma y (* -0.16666666666666666 (* y y)) y) y) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999) {
		tmp = sin(y) * (x / (z_m * y));
	} else {
		tmp = (fma(y, (-0.16666666666666666 * (y * y)), y) / y) * (x / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999)
		tmp = Float64(sin(y) * Float64(x / Float64(z_m * y)));
	else
		tmp = Float64(Float64(fma(y, Float64(-0.16666666666666666 * Float64(y * y)), y) / y) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.998999999999999999

   1. Initial program 90.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\sin y \cdot \frac{\frac{1}{y} \cdot x}{z}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{z} \cdot \sin y} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{y}}}{z} \cdot \sin y \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\frac{\color{blue}{x}}{y}}{z} \cdot \sin y \]

      12. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x}{z \cdot y}} \cdot \sin y \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{z \cdot y}} \cdot \sin y \]

      14. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot z}} \cdot \sin y \]

      15. lower-*.f64 92.6

          \[\leadsto \frac{x}{\color{blue}{y \cdot z}} \cdot \sin y \]

   4. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot z} \cdot \sin y} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 71.4

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)}}{z} \cdot \frac{x}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right) + y \cdot 1}}{z} \cdot \frac{x}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{6}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \frac{-1}{6}} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1}{z} \cdot \frac{x}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{z} \cdot \frac{x}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y \cdot {y}^{2}, y\right)}}{z} \cdot \frac{x}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right)}{z} \cdot \frac{x}{y} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

      10. lower-*.f64 71.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{z} \cdot \frac{x}{y} \]

   7. Applied rewrites 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}}{z} \cdot \frac{x}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \frac{x}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z}} \cdot \frac{x}{y} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{z} \cdot \color{blue}{\frac{x}{y}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{z \cdot y}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right) \cdot x}{\color{blue}{y \cdot z}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \color{blue}{\frac{x}{z}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]

      9. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}} \cdot \frac{x}{z} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 10^{-247}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z\_m}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (* x (/ (sin y) y)) 1e-247)
    (* (/ y z_m) (/ x y))
    (/ 1.0 (/ z_m x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((x * (sin(y) / y)) <= 1e-247) {
		tmp = (y / z_m) * (x / y);
	} else {
		tmp = 1.0 / (z_m / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * (sin(y) / y)) <= 1d-247) then
        tmp = (y / z_m) * (x / y)
    else
        tmp = 1.0d0 / (z_m / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((x * (Math.sin(y) / y)) <= 1e-247) {
		tmp = (y / z_m) * (x / y);
	} else {
		tmp = 1.0 / (z_m / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (x * (math.sin(y) / y)) <= 1e-247:
		tmp = (y / z_m) * (x / y)
	else:
		tmp = 1.0 / (z_m / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(x * Float64(sin(y) / y)) <= 1e-247)
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	else
		tmp = Float64(1.0 / Float64(z_m / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((x * (sin(y) / y)) <= 1e-247)
		tmp = (y / z_m) * (x / y);
	else
		tmp = 1.0 / (z_m / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1e-247], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 1e-247

   1. Initial program 91.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 78.9

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. lower-/.f64 46.6

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   7. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   if 1e-247 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 67.1

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 12.5)
    (/
     (*
      x
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       1.0))
     z_m)
    (* (/ y z_m) (/ x y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x * fma((y * y), fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / z_m;
	} else {
		tmp = (y / z_m) * (x / y);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 12.5)
		tmp = Float64(Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / z_m);
	else
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 12.5], N[(N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 12.5

   1. Initial program 97.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, 1\right)}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, 1\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, 1\right)}{z} \]

      5. sub-neg N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)}{z} \]

      7. associate-*l* N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right) + \color{blue}{\frac{-1}{6}}, 1\right)}{z} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right), \frac{-1}{6}\right)}, 1\right)}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}, \frac{-1}{6}\right), 1\right)}{z} \]

      11. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)}{z} \]

      12. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)}{z} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)}{z} \]

      14. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)}{z} \]

      15. lower-*.f64 69.0

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{z} \]

   5. Applied rewrites 69.0%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}{z} \]

   if 12.5 < y

   1. Initial program 89.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 89.2

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. lower-/.f64 26.2

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   7. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), x\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 4.5)
    (/
     (fma
      (* y y)
      (* x (fma (* y y) 0.008333333333333333 -0.16666666666666666))
      x)
     z_m)
    (* (/ y z_m) (/ x y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 4.5) {
		tmp = fma((y * y), (x * fma((y * y), 0.008333333333333333, -0.16666666666666666)), x) / z_m;
	} else {
		tmp = (y / z_m) * (x / y);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 4.5)
		tmp = Float64(fma(Float64(y * y), Float64(x * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), x) / z_m);
	else
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 4.5], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.5

   1. Initial program 97.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right), x\right)}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right), x\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right), x\right)}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \frac{-1}{6} \cdot x + \frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot x\right)}, x\right)}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \frac{-1}{6} \cdot x + \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right)}{z} \]

      7. distribute-rgt-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \color{blue}{x \cdot \left(\frac{-1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, x\right)}{z} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, x\right)}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), x\right)}{z} \]

      10. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, x\right)}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, x\right)}{z} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, x\right)}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), x\right)}{z} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \left({y}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right), x\right)}{z} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, x\right)}{z} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{-1}{6}\right), x\right)}{z} \]

      17. lower-*.f64 68.8

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, -0.16666666666666666\right), x\right)}{z} \]

   5. Applied rewrites 68.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), x\right)}}{z} \]

   if 4.5 < y

   1. Initial program 89.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 89.2

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. lower-/.f64 26.2

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   7. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(y \cdot y\right), x\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 12.5)
    (/ (fma x (* -0.16666666666666666 (* y y)) x) z_m)
    (* (/ y z_m) (/ x y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 12.5) {
		tmp = fma(x, (-0.16666666666666666 * (y * y)), x) / z_m;
	} else {
		tmp = (y / z_m) * (x / y);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 12.5)
		tmp = Float64(fma(x, Float64(-0.16666666666666666 * Float64(y * y)), x) / z_m);
	else
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 12.5], N[(N[(x * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 12.5

   1. Initial program 97.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)}}{z} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x + \color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}}}{z} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{-1}{6}\right)} \cdot {y}^{2} + x}{z} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)} + x}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {y}^{2}, x\right)}}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {y}^{2}}, x\right)}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)}, x\right)}{z} \]

      8. lower-*.f64 68.9

         \[\leadsto \frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}, x\right)}{z} \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(y \cdot y\right), x\right)}}{z} \]

   if 12.5 < y

   1. Initial program 89.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 89.2

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. lower-/.f64 26.2

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   7. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 12.5)
    (* (/ x z_m) (fma y (* y -0.16666666666666666) 1.0))
    (* (/ y z_m) (/ x y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x / z_m) * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = (y / z_m) * (x / y);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 12.5)
		tmp = Float64(Float64(x / z_m) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 12.5], N[(N[(x / z$95$m), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 12.5

   1. Initial program 97.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)}}{z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)}}{z} \]

      3. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right)}{z} \]

      4. lower-*.f64 68.9

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right)}{z} \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right)}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right)}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \cdot x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \cdot \frac{x}{z}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right)} \]

      7. lower-/.f64 70.3

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   7. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)} \]

   if 12.5 < y

   1. Initial program 89.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{\sin y}{y}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{\sin y}{y}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y} \cdot x}}{z} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin y}{y}} \cdot x}{z} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot \frac{1}{y}\right)} \cdot x}{z} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sin y \cdot \left(\frac{1}{y} \cdot x\right)}}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \left(\frac{1}{y} \cdot x\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{z}} \cdot \left(\frac{1}{y} \cdot x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{1 \cdot x}{y}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\sin y}{z} \cdot \frac{\color{blue}{x}}{y} \]

      12. lower-/.f64 89.2

          \[\leadsto \frac{\sin y}{z} \cdot \color{blue}{\frac{x}{y}} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\sin y}{z} \cdot \frac{x}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
   6. Step-by-step derivation

      1. lower-/.f64 26.2

         \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]

   7. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\frac{y}{z}} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \frac{1}{\frac{z\_m}{x}} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ 1.0 (/ z_m x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (1.0 / (z_m / x));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (1.0d0 / (z_m / x))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (1.0 / (z_m / x));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (1.0 / (z_m / x))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(1.0 / Float64(z_m / x)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (1.0 / (z_m / x));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{z}} \]
4. Step-by-step derivation

   1. lower-/.f64 59.1

      \[\leadsto \color{blue}{\frac{x}{z}} \]

5. Applied rewrites 59.1%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Step-by-step derivation

7. Applied rewrites 59.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
8. Add Preprocessing

Alternative 11: 58.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \frac{x}{z\_m} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ x z_m)))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / z_m);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (x / z_m)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / z_m);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (x / z_m)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(x / z_m))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (x / z_m);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{z}} \]
4. Step-by-step derivation

   1. lower-/.f64 59.1

      \[\leadsto \color{blue}{\frac{x}{z}} \]

5. Applied rewrites 59.1%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))

  (/ (* x (/ (sin y) y)) z))