Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.6%

Time: 10.6s

Alternatives: 10

Speedup: 0.9×

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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x\_m \cdot \frac{-1}{z}}{\frac{y}{\sin \left(-y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 6e-50)
    (/ (* x_m (/ -1.0 z)) (/ y (sin (- y))))
    (/ (* 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 <= 6e-50) {
		tmp = (x_m * (-1.0 / z)) / (y / sin(-y));
	} 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 <= 6d-50) then
        tmp = (x_m * ((-1.0d0) / z)) / (y / sin(-y))
    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 <= 6e-50) {
		tmp = (x_m * (-1.0 / z)) / (y / Math.sin(-y));
	} 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 <= 6e-50:
		tmp = (x_m * (-1.0 / z)) / (y / math.sin(-y))
	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 <= 6e-50)
		tmp = Float64(Float64(x_m * Float64(-1.0 / z)) / Float64(y / sin(Float64(-y))));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / 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 <= 6e-50)
		tmp = (x_m * (-1.0 / z)) / (y / sin(-y));
	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, 6e-50], N[(N[(x$95$m * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999981e-50

   1. Initial program 94.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sin-neg N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-lowering-neg.f64 97.6

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

   4. Applied egg-rr 97.6%

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

      1. distribute-rgt-neg-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. distribute-frac-neg2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. /-lowering-/.f64 97.6

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

   6. Applied egg-rr 97.6%

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

   if 5.99999999999999981e-50 < 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. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \frac{-1}{z}}{\frac{y}{\sin \left(-y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin y) y)))
   (*
    x_s
    (if (<= t_0 -2e-307)
      (/ (sin y) (* y (/ z x_m)))
      (if (<= t_0 2e-6) (* (/ (sin y) z) (/ x_m 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 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = sin(y) / (y * (z / x_m));
	} else if (t_0 <= 2e-6) {
		tmp = (sin(y) / z) * (x_m / 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 = sin(y) / y
    if (t_0 <= (-2d-307)) then
        tmp = sin(y) / (y * (z / x_m))
    else if (t_0 <= 2d-6) then
        tmp = (sin(y) / z) * (x_m / 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 = Math.sin(y) / y;
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} else if (t_0 <= 2e-6) {
		tmp = (Math.sin(y) / z) * (x_m / 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 = math.sin(y) / y
	tmp = 0
	if t_0 <= -2e-307:
		tmp = math.sin(y) / (y * (z / x_m))
	elif t_0 <= 2e-6:
		tmp = (math.sin(y) / z) * (x_m / 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(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	elseif (t_0 <= 2e-6)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / 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 = sin(y) / y;
	tmp = 0.0;
	if (t_0 <= -2e-307)
		tmp = sin(y) / (y * (z / x_m));
	elseif (t_0 <= 2e-6)
		tmp = (sin(y) / z) * (x_m / 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[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-307], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307

   1. Initial program 86.8%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 96.8

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

   4. Applied egg-rr 96.8%

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

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 1.99999999999999991e-6

   1. Initial program 96.3%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 96.1

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

   4. Applied egg-rr 96.1%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \frac{y \cdot \left(x\_m \cdot -0.16666666666666666\right)}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-320}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 -4e-122)
      (* y (/ (* y (* x_m -0.16666666666666666)) z))
      (if (<= t_0 5e-320) (* y (/ x_m (* 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 <= -4e-122) {
		tmp = y * ((y * (x_m * -0.16666666666666666)) / z);
	} else if (t_0 <= 5e-320) {
		tmp = y * (x_m / (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 <= (-4d-122)) then
        tmp = y * ((y * (x_m * (-0.16666666666666666d0))) / z)
    else if (t_0 <= 5d-320) then
        tmp = y * (x_m / (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 <= -4e-122) {
		tmp = y * ((y * (x_m * -0.16666666666666666)) / z);
	} else if (t_0 <= 5e-320) {
		tmp = y * (x_m / (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 <= -4e-122:
		tmp = y * ((y * (x_m * -0.16666666666666666)) / z)
	elif t_0 <= 5e-320:
		tmp = y * (x_m / (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 <= -4e-122)
		tmp = Float64(y * Float64(Float64(y * Float64(x_m * -0.16666666666666666)) / z));
	elseif (t_0 <= 5e-320)
		tmp = Float64(y * Float64(x_m / 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 <= -4e-122)
		tmp = y * ((y * (x_m * -0.16666666666666666)) / z);
	elseif (t_0 <= 5e-320)
		tmp = y * (x_m / (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, -4e-122], N[(y * N[(N[(y * N[(x$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-320], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $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) < -4.00000000000000024e-122

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 5.4

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

   8. Simplified 5.4%

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

   if -4.00000000000000024e-122 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.99994e-320

   1. Initial program 89.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 94.2

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 68.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 76.9

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

   9. Applied egg-rr 76.9%

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

   if 4.99994e-320 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 65.3

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

   5. Simplified 65.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -4 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \frac{y \cdot \left(x \cdot -0.16666666666666666\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 5 \cdot 10^{-320}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 0.5× speedup

Math

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

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 (* x_m (/ (sin y) y))))
   (* x_s (if (<= t_0 0.0) (/ (sin y) (* y (/ 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 = x_m * (sin(y) / y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sin(y) / (y * (z / x_m));
	} else {
		tmp = 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 = x_m * (sin(y) / y)
    if (t_0 <= 0.0d0) then
        tmp = sin(y) / (y * (z / x_m))
    else
        tmp = 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 = x_m * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} else {
		tmp = 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 = x_m * (math.sin(y) / y)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sin(y) / (y * (z / x_m))
	else:
		tmp = 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(x_m * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(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 = x_m * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sin(y) / (y * (z / x_m));
	else
		tmp = 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[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.5%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 90.8

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

   4. Applied egg-rr 90.8%

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

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

   1. Initial program 99.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 0:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.5× speedup

Math

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

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 (<= (/ (sin y) y) 2e-6) (* (sin y) (/ x_m (* 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 ((sin(y) / y) <= 2e-6) {
		tmp = sin(y) * (x_m / (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 ((sin(y) / y) <= 2d-6) then
        tmp = sin(y) * (x_m / (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 ((Math.sin(y) / y) <= 2e-6) {
		tmp = Math.sin(y) * (x_m / (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 (math.sin(y) / y) <= 2e-6:
		tmp = math.sin(y) * (x_m / (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(sin(y) / y) <= 2e-6)
		tmp = Float64(sin(y) * Float64(x_m / 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 ((sin(y) / y) <= 2e-6)
		tmp = sin(y) * (x_m / (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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 2e-6], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.99999999999999991e-6

   1. Initial program 91.2%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sin-lowering-sin.f64 91.7

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

   4. Applied egg-rr 91.7%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -4 \cdot 10^{-100}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{-1}{z}}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, -1\right)}\\ \end{array} \end{array} \]

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) -4e-100)
    (/ (* x_m (fma -0.16666666666666666 (* y y) 1.0)) z)
    (/ (* x_m (/ -1.0 z)) (fma -0.16666666666666666 (* y y) -1.0)))))

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) <= -4e-100) {
		tmp = (x_m * fma(-0.16666666666666666, (y * y), 1.0)) / z;
	} else {
		tmp = (x_m * (-1.0 / z)) / fma(-0.16666666666666666, (y * y), -1.0);
	}
	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) <= -4e-100)
		tmp = Float64(Float64(x_m * fma(-0.16666666666666666, Float64(y * y), 1.0)) / z);
	else
		tmp = Float64(Float64(x_m * Float64(-1.0 / z)) / fma(-0.16666666666666666, Float64(y * y), -1.0));
	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[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -4e-100], N[(N[(x$95$m * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.0000000000000001e-100

   1. Initial program 99.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

   if -4.0000000000000001e-100 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sin-neg N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-lowering-neg.f64 98.2

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

   4. Applied egg-rr 98.2%

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

      1. distribute-rgt-neg-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. distribute-frac-neg2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. /-lowering-/.f64 98.2

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.9

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

   9. Simplified 70.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -4 \cdot 10^{-100}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1}{z}}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-106}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 1e-106) (* y (/ x_m (* 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 ((sin(y) / y) <= 1e-106) {
		tmp = y * (x_m / (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 ((sin(y) / y) <= 1d-106) then
        tmp = y * (x_m / (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 ((Math.sin(y) / y) <= 1e-106) {
		tmp = y * (x_m / (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 (math.sin(y) / y) <= 1e-106:
		tmp = y * (x_m / (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(sin(y) / y) <= 1e-106)
		tmp = Float64(y * Float64(x_m / 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 ((sin(y) / y) <= 1e-106)
		tmp = y * (x_m / (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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-106], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.99999999999999941e-107

   1. Initial program 90.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.8

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 30.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 38.2

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

   9. Applied egg-rr 38.2%

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

   if 9.99999999999999941e-107 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 88.9

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

   5. Simplified 88.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-106}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

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 1.05e+50)
    (/ (* x_m (fma -0.16666666666666666 (* y y) 1.0)) z)
    (* y (/ x_m (* 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 <= 1.05e+50) {
		tmp = (x_m * fma(-0.16666666666666666, (y * y), 1.0)) / z;
	} else {
		tmp = y * (x_m / (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 <= 1.05e+50)
		tmp = Float64(Float64(x_m * fma(-0.16666666666666666, Float64(y * y), 1.0)) / z);
	else
		tmp = Float64(y * Float64(x_m / 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, 1.05e+50], 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[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e50

   1. Initial program 96.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. accelerator-lowering-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. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   if 1.05e50 < y

   1. Initial program 92.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 87.8

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 29.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 38.3

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

   9. Applied egg-rr 38.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+50}:\\ \;\;\;\;x\_m \cdot \frac{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

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 1.52e+50)
    (* x_m (/ (fma y (* y -0.16666666666666666) 1.0) z))
    (* y (/ x_m (* 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 <= 1.52e+50) {
		tmp = x_m * (fma(y, (y * -0.16666666666666666), 1.0) / z);
	} else {
		tmp = y * (x_m / (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 <= 1.52e+50)
		tmp = Float64(x_m * Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) / z));
	else
		tmp = Float64(y * Float64(x_m / 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, 1.52e+50], N[(x$95$m * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.5199999999999999e50

   1. Initial program 96.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 90.0

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

   4. Applied egg-rr 90.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \frac{{y}^{2}}{z} + \frac{1}{z}\right)} \cdot x \]
   6. 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. *-rgt-identity N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 69.8

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

   7. Simplified 69.8%

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

   if 1.5199999999999999e50 < y

   1. Initial program 92.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sin-lowering-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 87.8

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 29.2%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 38.3

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

   9. Applied egg-rr 38.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% 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 95.8%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 63.0

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

5. Simplified 63.0%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024199 
(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))