Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.5% → 98.4%

Time: 6.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.5% 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: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\sin y}{z}}{y} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 3e-133)
    (* (/ (/ (sin y) z) y) x_m)
    (/ (* 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 <= 3e-133) {
		tmp = ((sin(y) / z) / y) * x_m;
	} 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 <= 3d-133) then
        tmp = ((sin(y) / z) / y) * x_m
    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 <= 3e-133) {
		tmp = ((Math.sin(y) / z) / y) * x_m;
	} 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 <= 3e-133:
		tmp = ((math.sin(y) / z) / y) * x_m
	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 <= 3e-133)
		tmp = Float64(Float64(Float64(sin(y) / z) / y) * x_m);
	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 <= 3e-133)
		tmp = ((sin(y) / z) / y) * x_m;
	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, 3e-133], N[(N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision] * x$95$m), $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 < 3.00000000000000019e-133

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. distribute-frac-neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 3.00000000000000019e-133 < 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 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\sin y}{z}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 2e-314) (/ (/ (* (sin y) x_m) z) y) (/ 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 <= 2e-314) {
		tmp = ((sin(y) * x_m) / z) / y;
	} 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 <= 2d-314) then
        tmp = ((sin(y) * x_m) / z) / y
    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 <= 2e-314) {
		tmp = ((Math.sin(y) * x_m) / z) / y;
	} 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 <= 2e-314:
		tmp = ((math.sin(y) * x_m) / z) / y
	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 <= 2e-314)
		tmp = Float64(Float64(Float64(sin(y) * x_m) / z) / y);
	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 <= 2e-314)
		tmp = ((sin(y) * x_m) / z) / y;
	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, 2e-314], N[(N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 1.9999999999e-314

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-commutative N/A

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

      12. frac-2neg-rev N/A

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

      13. remove-double-neg N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. remove-double-neg N/A

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

      19. lower-*.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 1.9999999999e-314 < (*.f64 x (/.f64 (sin.f64 y) y))

   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 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-314}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 2e-314) (/ (* x_m (/ (sin y) z)) y) (/ 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 <= 2e-314) {
		tmp = (x_m * (sin(y) / z)) / y;
	} 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 <= 2d-314) then
        tmp = (x_m * (sin(y) / z)) / y
    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 <= 2e-314) {
		tmp = (x_m * (Math.sin(y) / z)) / y;
	} 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 <= 2e-314:
		tmp = (x_m * (math.sin(y) / z)) / y
	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 <= 2e-314)
		tmp = Float64(Float64(x_m * Float64(sin(y) / z)) / y);
	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 <= 2e-314)
		tmp = (x_m * (sin(y) / z)) / y;
	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, 2e-314], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 1.9999999999e-314

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 1.9999999999e-314 < (*.f64 x (/.f64 (sin.f64 y) y))

   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 4: 95.2% accurate, 0.5× speedup

Math

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

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) 0.995) (* (/ (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 tmp;
	if ((sin(y) / y) <= 0.995) {
		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) :: tmp
    if ((sin(y) / y) <= 0.995d0) 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 tmp;
	if ((Math.sin(y) / y) <= 0.995) {
		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):
	tmp = 0
	if (math.sin(y) / y) <= 0.995:
		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)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.995)
		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)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.995)
		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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.995], 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 2 regimes

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

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 95.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.995) {
		tmp = sin(y) * (x_m / (z * y));
	} else {
		tmp = x_m / 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) <= 0.995d0) 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) <= 0.995) {
		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) <= 0.995:
		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) <= 0.995)
		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) <= 0.995)
		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], 0.995], 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) < 0.994999999999999996

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-commutative N/A

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

      12. frac-2neg-rev N/A

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

      13. remove-double-neg N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. remove-double-neg N/A

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

      19. lower-*.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

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

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

      19. remove-double-neg N/A

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

      20. lift-*.f64 90.2

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

   6. Applied rewrites 90.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 55.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 7: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 4e-86) (* (- y) (/ x_m (* (- y) z))) (/ 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) <= 4e-86) {
		tmp = -y * (x_m / (-y * z));
	} 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) <= 4d-86) then
        tmp = -y * (x_m / (-y * z))
    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) <= 4e-86) {
		tmp = -y * (x_m / (-y * z));
	} 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) <= 4e-86:
		tmp = -y * (x_m / (-y * z))
	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) <= 4e-86)
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(Float64(-y) * z)));
	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) <= 4e-86)
		tmp = -y * (x_m / (-y * z));
	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], 4e-86], N[((-y) * N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 4.00000000000000034e-86

   1. Initial program 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.6

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

   7. Applied rewrites 43.6%

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

   if 4.00000000000000034e-86 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

Alternative 8: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sin y}{y \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

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 3e-133)
    (* (/ (sin y) (* y z)) x_m)
    (/ (* 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 <= 3e-133) {
		tmp = (sin(y) / (y * z)) * x_m;
	} 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 <= 3d-133) then
        tmp = (sin(y) / (y * z)) * x_m
    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 <= 3e-133) {
		tmp = (Math.sin(y) / (y * z)) * x_m;
	} 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 <= 3e-133:
		tmp = (math.sin(y) / (y * z)) * x_m
	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 <= 3e-133)
		tmp = Float64(Float64(sin(y) / Float64(y * z)) * x_m);
	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 <= 3e-133)
		tmp = (sin(y) / (y * z)) * x_m;
	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, 3e-133], N[(N[(N[Sin[y], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $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 < 3.00000000000000019e-133

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. distribute-frac-neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 88.0

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

   6. Applied rewrites 88.0%

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

   if 3.00000000000000019e-133 < 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 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sin y}{y \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot x\_m, y \cdot y, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \end{array} \end{array} \]

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 1000000.0)
    (/ (fma (* -0.16666666666666666 x_m) (* y y) 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 <= 1000000.0) {
		tmp = fma((-0.16666666666666666 * x_m), (y * y), 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 <= 1000000.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x_m), Float64(y * y), x_m) / z);
	else
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(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, 1000000.0], N[(N[(N[(-0.16666666666666666 * x$95$m), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $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 < 1e6

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.9%

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

   if 1e6 < y

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.8

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

   7. Applied rewrites 46.8%

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

Alternative 10: 60.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1000000:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \end{array} \end{array} \]

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 1000000.0)
    (* (/ 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 <= 1000000.0) {
		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 <= 1000000.0)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(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, 1000000.0], 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 < 1e6

   1. Initial program 98.2%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.8

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

   7. Applied rewrites 69.8%

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

   if 1e6 < y

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.8

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

   7. Applied rewrites 46.8%

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

Alternative 11: 58.4% 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 96.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 62.3

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

5. Applied rewrites 62.3%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024326 
(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))