Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.8%

Time: 10.9s

Alternatives: 8

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: 96.0% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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-11) (* 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-11) {
		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-11) 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-11) {
		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-11:
		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-11)
		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-11)
		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-11], 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 < 1.99999999999999988e-11

   1. Initial program 96.4%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.4

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

   4. Applied egg-rr 96.4%

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

   if 1.99999999999999988e-11 < x

   1. Initial program 99.8%

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

Alternative 2: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \end{array} \]

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 4.2e+169)
    (* (/ (sin y) y) (/ x_m z))
    (* x_m (/ (sin y) (* y 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 <= 4.2e+169) {
		tmp = (sin(y) / y) * (x_m / z);
	} else {
		tmp = x_m * (sin(y) / (y * 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 <= 4.2d+169) then
        tmp = (sin(y) / y) * (x_m / z)
    else
        tmp = x_m * (sin(y) / (y * 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 <= 4.2e+169) {
		tmp = (Math.sin(y) / y) * (x_m / z);
	} else {
		tmp = x_m * (Math.sin(y) / (y * 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 <= 4.2e+169:
		tmp = (math.sin(y) / y) * (x_m / z)
	else:
		tmp = x_m * (math.sin(y) / (y * 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 (x_m <= 4.2e+169)
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	else
		tmp = Float64(x_m * Float64(sin(y) / Float64(y * 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 <= 4.2e+169)
		tmp = (sin(y) / y) * (x_m / z);
	else
		tmp = x_m * (sin(y) / (y * 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[x$95$m, 4.2e+169], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000002e169

   1. Initial program 96.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   if 4.2000000000000002e169 < x

   1. Initial program 99.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.4

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

   4. Applied egg-rr 96.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 2e-6)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* (sin y) (/ x_m (* y 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 (y <= 2e-6) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = sin(y) * (x_m / (y * 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 (y <= 2e-6)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * 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[y, 2e-6], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999991e-6

   1. Initial program 97.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.4

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

   4. Applied egg-rr 96.4%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.9

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

   7. Simplified 76.9%

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

   if 1.99999999999999991e-6 < y

   1. Initial program 95.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 94.1

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

   4. Applied egg-rr 94.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 6.4)
    (/
     (*
      x_m
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) -0.0001984126984126984 0.008333333333333333)
         -0.16666666666666666))
       1.0))
     z)
    (* y (/ x_m (* y 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 (y <= 6.4) {
		tmp = (x_m * fma(y, (y * fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), 1.0)) / z;
	} else {
		tmp = y * (x_m / (y * 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 (y <= 6.4)
		tmp = Float64(Float64(x_m * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), 1.0)) / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(y * 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[y, 6.4], N[(N[(x$95$m * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000004

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 74.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}}{z} \]
   7. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.9

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

   8. Simplified 74.9%

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

   if 6.4000000000000004 < y

   1. Initial program 95.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.0

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 38.6

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

   9. Applied egg-rr 38.6%

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

Alternative 5: 59.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 6.4)
    (/ (* x_m (fma -0.16666666666666666 (* y y) 1.0)) z)
    (* y (/ x_m (* y 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 (y <= 6.4) {
		tmp = (x_m * fma(-0.16666666666666666, (y * y), 1.0)) / z;
	} else {
		tmp = y * (x_m / (y * 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 (y <= 6.4)
		tmp = Float64(Float64(x_m * fma(-0.16666666666666666, Float64(y * y), 1.0)) / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(y * 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[y, 6.4], N[(N[(x$95$m * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000004

   1. Initial program 97.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 75.1

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

   5. Simplified 75.1%

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

   if 6.4000000000000004 < y

   1. Initial program 95.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.0

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 38.6

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

   9. Applied egg-rr 38.6%

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

Alternative 6: 60.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 6.4)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* y (/ x_m (* y 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 (y <= 6.4) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = y * (x_m / (y * 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 (y <= 6.4)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(y * Float64(x_m / Float64(y * 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[y, 6.4], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000004

   1. Initial program 97.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.0

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

   7. Simplified 77.0%

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

   if 6.4000000000000004 < y

   1. Initial program 95.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.0

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 28.0

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

   7. Simplified 28.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 38.6

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

   9. Applied egg-rr 38.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y 1e-18) (/ x_m z) (* y (/ x_m (* y 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 (y <= 1e-18) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (y * 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 (y <= 1d-18) then
        tmp = x_m / z
    else
        tmp = y * (x_m / (y * 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 (y <= 1e-18) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (y * 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 y <= 1e-18:
		tmp = x_m / z
	else:
		tmp = y * (x_m / (y * 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 (y <= 1e-18)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(y * 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 (y <= 1e-18)
		tmp = x_m / z;
	else
		tmp = y * (x_m / (y * 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[y, 1e-18], N[(x$95$m / z), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.0000000000000001e-18

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 79.5

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

   5. Simplified 79.5%

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

   if 1.0000000000000001e-18 < y

   1. Initial program 95.5%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.1

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

   4. Applied egg-rr 94.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 29.9

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

   7. Simplified 29.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 40.2

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

   9. Applied egg-rr 40.2%

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

Alternative 8: 58.2% 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 97.3%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 64.4

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

5. Simplified 64.4%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024208 
(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))