Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.7%

Time: 7.7s

Alternatives: 12

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: 96.1% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+37}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 2e+37) (* t_0 (/ x_m z)) (/ (* x_m t_0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 2e+37) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 2d+37) then
        tmp = t_0 * (x_m / z)
    else
        tmp = (x_m * t_0) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 2e+37) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 2e+37:
		tmp = t_0 * (x_m / z)
	else:
		tmp = (x_m * t_0) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 2e+37)
		tmp = Float64(t_0 * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * t_0) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 2e+37)
		tmp = t_0 * (x_m / z);
	else
		tmp = (x_m * t_0) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2e+37], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999991e37

   1. Initial program 94.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   if 1.99999999999999991e37 < x

   1. Initial program 99.7%

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

Alternative 2: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\_m\right)}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -5e-177)
      (* (fma (* y y) -0.16666666666666666 1.0) (/ x_m z))
      (if (<= t_0 0.0)
        (/ (* x_m y) (* z y))
        (/
         (fma
          (* x_m (fma (* 0.008333333333333333 y) y -0.16666666666666666))
          (* y y)
          x_m)
         z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-177) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x_m / z);
	} else if (t_0 <= 0.0) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = fma((x_m * fma((0.008333333333333333 * y), y, -0.16666666666666666)), (y * y), x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -5e-177)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x_m / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	else
		tmp = Float64(fma(Float64(x_m * fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666)), Float64(y * y), x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-177], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5e-177

   1. Initial program 99.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.5

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.1

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

   7. Applied rewrites 64.1%

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

   if -5e-177 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -0.0

   1. Initial program 86.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 56.7

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

   7. Applied rewrites 56.7%

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

   if -0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\right)}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\frac{x \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), y \cdot y, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(y \cdot y\right) - 1\\ t_1 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{t\_0}{t\_0 \cdot z}\right) \cdot x\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(-0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot y, y, \left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666\right), y \cdot y, 1\right)}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- (* -0.16666666666666666 (* y y)) 1.0)) (t_1 (/ (sin y) y)))
   (*
    x_s
    (if (<= t_1 -4e-154)
      (* (* (fma -0.16666666666666666 (* y y) 1.0) (/ t_0 (* t_0 z))) x_m)
      (if (<= t_1 5e-7)
        (* (/ y z) (/ x_m y))
        (/
         (*
          x_m
          (fma
           (fma
            (* (* -0.0001984126984126984 (* y y)) y)
            y
            (- (* (* y y) 0.008333333333333333) 0.16666666666666666))
           (* y y)
           1.0))
         z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (-0.16666666666666666 * (y * y)) - 1.0;
	double t_1 = sin(y) / y;
	double tmp;
	if (t_1 <= -4e-154) {
		tmp = (fma(-0.16666666666666666, (y * y), 1.0) * (t_0 / (t_0 * z))) * x_m;
	} else if (t_1 <= 5e-7) {
		tmp = (y / z) * (x_m / y);
	} else {
		tmp = (x_m * fma(fma(((-0.0001984126984126984 * (y * y)) * y), y, (((y * y) * 0.008333333333333333) - 0.16666666666666666)), (y * y), 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(-0.16666666666666666 * Float64(y * y)) - 1.0)
	t_1 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_1 <= -4e-154)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(t_0 / Float64(t_0 * z))) * x_m);
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(y / z) * Float64(x_m / y));
	else
		tmp = Float64(Float64(x_m * fma(fma(Float64(Float64(-0.0001984126984126984 * Float64(y * y)) * y), y, Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666)), Float64(y * y), 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -4e-154], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$0 / N[(t$95$0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y + N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -3.9999999999999999e-154

   1. Initial program 96.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 89.1

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

   6. Applied rewrites 89.1%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 18.9

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

   9. Applied rewrites 18.9%

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

   11. Applied rewrites 34.6%

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

   if -3.9999999999999999e-154 < (/.f64 (sin.f64 y) y) < 4.99999999999999977e-7

   1. Initial program 89.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.0

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

   4. Applied rewrites 89.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 14.4

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

   7. Applied rewrites 14.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.8

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

   9. Applied rewrites 21.8%

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

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({y}^{4}, -0.0001984126984126984, 0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(-0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot y, y, \left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666\right), \color{blue}{y} \cdot y, 1\right)}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-177}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -5e-177)
      (* (* (/ (* y y) z) -0.16666666666666666) x_m)
      (if (<= t_0 0.0) (/ (* x_m y) (* z y)) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-177) {
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	} else if (t_0 <= 0.0) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-5d-177)) then
        tmp = (((y * y) / z) * (-0.16666666666666666d0)) * x_m
    else if (t_0 <= 0.0d0) then
        tmp = (x_m * y) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-177) {
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	} else if (t_0 <= 0.0) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -5e-177:
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m
	elif t_0 <= 0.0:
		tmp = (x_m * y) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -5e-177)
		tmp = Float64(Float64(Float64(Float64(y * y) / z) * -0.16666666666666666) * x_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -5e-177)
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	elseif (t_0 <= 0.0)
		tmp = (x_m * y) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-177], N[(N[(N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5e-177

   1. Initial program 99.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 91.4

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

   6. Applied rewrites 91.4%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.0

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

   9. Applied rewrites 64.0%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 4.3%

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

   if -5e-177 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -0.0

   1. Initial program 86.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 56.7

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

   7. Applied rewrites 56.7%

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

   if -0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   5. Applied rewrites 58.3%

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

Alternative 5: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.995:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) - 0.16666666666666666, y \cdot y, 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.995)
    (/ (* (sin y) x_m) (* z y))
    (/
     (*
      x_m
      (fma
       (-
        (* (* y y) (fma (* -0.0001984126984126984 y) y 0.008333333333333333))
        0.16666666666666666)
       (* y y)
       1.0))
     z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.995) {
		tmp = (sin(y) * x_m) / (z * y);
	} else {
		tmp = (x_m * fma((((y * y) * fma((-0.0001984126984126984 * y), y, 0.008333333333333333)) - 0.16666666666666666), (y * y), 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.995)
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	else
		tmp = Float64(Float64(x_m * fma(Float64(Float64(Float64(y * y) * fma(Float64(-0.0001984126984126984 * y), y, 0.008333333333333333)) - 0.16666666666666666), Float64(y * y), 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.995], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(-0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.994999999999999996

   1. Initial program 91.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.9

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

   4. Applied rewrites 90.9%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({y}^{4}, -0.0001984126984126984, 0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) - 0.16666666666666666, \color{blue}{y} \cdot y, 1\right)}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.995:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) - 0.16666666666666666, y \cdot y, 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.995)
    (* (sin y) (/ x_m (* z y)))
    (/
     (*
      x_m
      (fma
       (-
        (* (* y y) (fma (* -0.0001984126984126984 y) y 0.008333333333333333))
        0.16666666666666666)
       (* y y)
       1.0))
     z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.995) {
		tmp = sin(y) * (x_m / (z * y));
	} else {
		tmp = (x_m * fma((((y * y) * fma((-0.0001984126984126984 * y), y, 0.008333333333333333)) - 0.16666666666666666), (y * y), 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.995)
		tmp = Float64(sin(y) * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(Float64(x_m * fma(Float64(Float64(Float64(y * y) * fma(Float64(-0.0001984126984126984 * y), y, 0.008333333333333333)) - 0.16666666666666666), Float64(y * y), 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.995], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(-0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.994999999999999996

   1. Initial program 91.3%

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

      1. remove-double-neg N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. remove-double-neg 90.8

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

   6. Applied rewrites 90.8%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({y}^{4}, -0.0001984126984126984, 0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) - 0.16666666666666666, \color{blue}{y} \cdot y, 1\right)}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 0:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0) (/ (* x_m y) (* z y)) (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 0.0) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 0.0d0) then
        tmp = (x_m * y) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 0.0) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 0.0:
		tmp = (x_m * y) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 0.0)
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= 0.0)
		tmp = (x_m * y) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -0.0

   1. Initial program 93.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

   if -0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.3

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

   5. Applied rewrites 58.3%

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

Alternative 8: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sin y}{z}}{y} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 2.5e-23) (* (/ (/ (sin y) z) y) x_m) (* (/ (sin y) y) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 2.5e-23) {
		tmp = ((sin(y) / z) / y) * x_m;
	} else {
		tmp = (sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.5d-23) then
        tmp = ((sin(y) / z) / y) * x_m
    else
        tmp = (sin(y) / y) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 2.5e-23) {
		tmp = ((Math.sin(y) / z) / y) * x_m;
	} else {
		tmp = (Math.sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= 2.5e-23:
		tmp = ((math.sin(y) / z) / y) * x_m
	else:
		tmp = (math.sin(y) / y) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 2.5e-23)
		tmp = Float64(Float64(Float64(sin(y) / z) / y) * x_m);
	else
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= 2.5e-23)
		tmp = ((sin(y) / z) / y) * x_m;
	else
		tmp = (sin(y) / y) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, 2.5e-23], N[(N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5000000000000001e-23

   1. Initial program 94.3%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 92.0

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

   6. Applied rewrites 92.0%

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

   if 2.5000000000000001e-23 < z

   1. Initial program 99.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 9: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
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. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.8

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

4. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
5. Add Preprocessing

Alternative 10: 56.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.8e+43)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (/ (* x_m y) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.8e+43) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = (x_m * y) / (z * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.8e+43)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.8e+43], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000005e43

   1. Initial program 97.7%

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

      1. remove-double-neg N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

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

      12. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if 1.80000000000000005e43 < y

   1. Initial program 88.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 16.8

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

   7. Applied rewrites 16.8%

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

Alternative 11: 57.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.2e+43)
    (* (fma (* y y) -0.16666666666666666 1.0) (/ x_m z))
    (/ (* x_m y) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.2e+43) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x_m / z);
	} else {
		tmp = (x_m * y) / (z * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.2e+43)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.2e+43], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.20000000000000014e43

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if 3.20000000000000014e43 < y

   1. Initial program 88.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 16.8

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

   7. Applied rewrites 16.8%

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

Alternative 12: 57.8% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 57.2

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

5. Applied rewrites 57.2%

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

Developer Target 1: 99.6% 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 2024327 
(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))