Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.7%

Time: 7.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 20000:\\ \;\;\;\;\frac{x}{\frac{z\_m}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 20000.0)
    (/ x (* (/ z_m (sin y)) y))
    (/ (* (/ (sin y) y) x) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 20000.0) {
		tmp = x / ((z_m / sin(y)) * y);
	} else {
		tmp = ((sin(y) / y) * x) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 20000.0d0) then
        tmp = x / ((z_m / sin(y)) * y)
    else
        tmp = ((sin(y) / y) * x) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 20000.0) {
		tmp = x / ((z_m / Math.sin(y)) * y);
	} else {
		tmp = ((Math.sin(y) / y) * x) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 20000.0:
		tmp = x / ((z_m / math.sin(y)) * y)
	else:
		tmp = ((math.sin(y) / y) * x) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 20000.0)
		tmp = Float64(x / Float64(Float64(z_m / sin(y)) * y));
	else
		tmp = Float64(Float64(Float64(sin(y) / y) * x) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 20000.0)
		tmp = x / ((z_m / sin(y)) * y);
	else
		tmp = ((sin(y) / y) * x) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 20000.0], N[(x / N[(N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e4

   1. Initial program 96.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if 2e4 < z

   1. Initial program 99.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 20000:\\ \;\;\;\;\frac{x}{\frac{z}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 41.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{y \cdot y}{z\_m} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y \cdot x}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x) z_m)))
   (*
    z_s
    (if (<= t_0 -1e-213)
      (* (* (/ (* y y) z_m) -0.16666666666666666) x)
      (if (<= t_0 0.0) (/ (* y x) (* y z_m)) (/ x z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = ((sin(y) / y) * x) / z_m;
	double tmp;
	if (t_0 <= -1e-213) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 0.0) {
		tmp = (y * x) / (y * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((sin(y) / y) * x) / z_m
    if (t_0 <= (-1d-213)) then
        tmp = (((y * y) / z_m) * (-0.16666666666666666d0)) * x
    else if (t_0 <= 0.0d0) then
        tmp = (y * x) / (y * z_m)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = ((Math.sin(y) / y) * x) / z_m;
	double tmp;
	if (t_0 <= -1e-213) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 0.0) {
		tmp = (y * x) / (y * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = ((math.sin(y) / y) * x) / z_m
	tmp = 0
	if t_0 <= -1e-213:
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x
	elif t_0 <= 0.0:
		tmp = (y * x) / (y * z_m)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x) / z_m)
	tmp = 0.0
	if (t_0 <= -1e-213)
		tmp = Float64(Float64(Float64(Float64(y * y) / z_m) * -0.16666666666666666) * x);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(y * x) / Float64(y * z_m));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = ((sin(y) / y) * x) / z_m;
	tmp = 0.0;
	if (t_0 <= -1e-213)
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	elseif (t_0 <= 0.0)
		tmp = (y * x) / (y * z_m);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, -1e-213], N[(N[(N[(N[(y * y), $MachinePrecision] / z$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(y * x), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.9999999999999995e-214

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 4.4%

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

   if -9.9999999999999995e-214 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 89.0%

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

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   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 69.8

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

   7. Applied rewrites 69.8%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 59.9

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

   5. Applied rewrites 59.9%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot x\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{y \cdot y}{z\_m} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{-311}:\\ \;\;\;\;\frac{x}{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) x)))
   (*
    z_s
    (if (<= t_0 -5e-86)
      (* (* (/ (* y y) z_m) -0.16666666666666666) x)
      (if (<= t_0 1e-311)
        (/ x (* (* (* y y) 0.16666666666666666) z_m))
        (/ x z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = (sin(y) / y) * x;
	double tmp;
	if (t_0 <= -5e-86) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 1e-311) {
		tmp = x / (((y * y) * 0.16666666666666666) * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(y) / y) * x
    if (t_0 <= (-5d-86)) then
        tmp = (((y * y) / z_m) * (-0.16666666666666666d0)) * x
    else if (t_0 <= 1d-311) then
        tmp = x / (((y * y) * 0.16666666666666666d0) * z_m)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = (Math.sin(y) / y) * x;
	double tmp;
	if (t_0 <= -5e-86) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else if (t_0 <= 1e-311) {
		tmp = x / (((y * y) * 0.16666666666666666) * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = (math.sin(y) / y) * x
	tmp = 0
	if t_0 <= -5e-86:
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x
	elif t_0 <= 1e-311:
		tmp = x / (((y * y) * 0.16666666666666666) * z_m)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(Float64(sin(y) / y) * x)
	tmp = 0.0
	if (t_0 <= -5e-86)
		tmp = Float64(Float64(Float64(Float64(y * y) / z_m) * -0.16666666666666666) * x);
	elseif (t_0 <= 1e-311)
		tmp = Float64(x / Float64(Float64(Float64(y * y) * 0.16666666666666666) * z_m));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = (sin(y) / y) * x;
	tmp = 0.0;
	if (t_0 <= -5e-86)
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	elseif (t_0 <= 1e-311)
		tmp = x / (((y * y) * 0.16666666666666666) * z_m);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, -5e-86], N[(N[(N[(N[(y * y), $MachinePrecision] / z$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 1e-311], N[(x / N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -4.9999999999999999e-86

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 8.5%

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

   if -4.9999999999999999e-86 < (*.f64 x (/.f64 (sin.f64 y) y)) < 9.99999999999948e-312

   1. Initial program 88.0%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 42.6%

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

   if 9.99999999999948e-312 < (*.f64 x (/.f64 (sin.f64 y) y))

   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 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot x \leq -5 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot x \leq 10^{-311}:\\ \;\;\;\;\frac{x}{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.8:\\ \;\;\;\;\frac{x}{y \cdot z\_m} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right) \cdot z\_m, y \cdot y, z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.8)
    (* (/ x (* y z_m)) (sin y))
    (/
     x
     (fma
      (* (fma 0.019444444444444445 (* y y) 0.16666666666666666) z_m)
      (* y y)
      z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.8) {
		tmp = (x / (y * z_m)) * sin(y);
	} else {
		tmp = x / fma((fma(0.019444444444444445, (y * y), 0.16666666666666666) * z_m), (y * y), z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.8)
		tmp = Float64(Float64(x / Float64(y * z_m)) * sin(y));
	else
		tmp = Float64(x / fma(Float64(fma(0.019444444444444445, Float64(y * y), 0.16666666666666666) * z_m), Float64(y * y), z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.8], N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(0.019444444444444445 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * z$95$m), $MachinePrecision] * N[(y * y), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.80000000000000004

   1. Initial program 94.0%

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 87.8%

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

   if 0.80000000000000004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.6

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

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{z + {y}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right) + z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right) \cdot {y}^{2}} + z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z, {y}^{2}, z\right)}} \]

   9. Applied rewrites 100.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.8:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right) \cdot z, y \cdot y, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z\_m} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{y \cdot y}{z\_m} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot z\_m\right), z\_m\right)}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (/ (sin y) y) x) z_m) -5e-45)
    (* (* (/ (* y y) z_m) -0.16666666666666666) x)
    (/ x (fma y (* 0.16666666666666666 (* y z_m)) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((sin(y) / y) * x) / z_m) <= -5e-45) {
		tmp = (((y * y) / z_m) * -0.16666666666666666) * x;
	} else {
		tmp = x / fma(y, (0.16666666666666666 * (y * z_m)), z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x) / z_m) <= -5e-45)
		tmp = Float64(Float64(Float64(Float64(y * y) / z_m) * -0.16666666666666666) * x);
	else
		tmp = Float64(x / fma(y, Float64(0.16666666666666666 * Float64(y * z_m)), z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], -5e-45], N[(N[(N[(N[(y * y), $MachinePrecision] / z$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision], N[(x / N[(y * N[(0.16666666666666666 * N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.99999999999999976e-45

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 5.0%

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

   if -4.99999999999999976e-45 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 85.3

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

   6. Applied rewrites 85.3%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 64.6

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

   9. Applied rewrites 64.6%

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

   11. Applied rewrites 64.6%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot z\right), z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], 0.0], N[(N[(y * x), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $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 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 56.4

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

   7. Applied rewrites 56.4%

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

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

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 59.9

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

   5. Applied rewrites 59.9%

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

4. Final simplification 57.9%

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

Alternative 7: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{\frac{z\_m}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{\frac{z\_m}{x} \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 8.5e+55)
    (/ x (* (/ z_m (sin y)) y))
    (/ (sin y) (* (/ z_m x) y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 8.5e+55) {
		tmp = x / ((z_m / sin(y)) * y);
	} else {
		tmp = sin(y) / ((z_m / x) * y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 8.5d+55) then
        tmp = x / ((z_m / sin(y)) * y)
    else
        tmp = sin(y) / ((z_m / x) * y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 8.5e+55) {
		tmp = x / ((z_m / Math.sin(y)) * y);
	} else {
		tmp = Math.sin(y) / ((z_m / x) * y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 8.5e+55:
		tmp = x / ((z_m / math.sin(y)) * y)
	else:
		tmp = math.sin(y) / ((z_m / x) * y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 8.5e+55)
		tmp = Float64(x / Float64(Float64(z_m / sin(y)) * y));
	else
		tmp = Float64(sin(y) / Float64(Float64(z_m / x) * y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 8.5e+55)
		tmp = x / ((z_m / sin(y)) * y);
	else
		tmp = sin(y) / ((z_m / x) * y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 8.5e+55], N[(x / N[(N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] / N[(N[(z$95$m / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.50000000000000002e55

   1. Initial program 96.5%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 91.4

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

   6. Applied rewrites 91.4%

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

   if 8.50000000000000002e55 < z

   1. Initial program 99.9%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. remove-double-div N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. div-inv N/A

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

      12. remove-double-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

Alternative 8: 81.9% accurate, 1.0× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 4e-5)
    (/ x (fma y (* 0.16666666666666666 (* y z_m)) z_m))
    (* (/ x y) (/ (sin y) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 4e-5) {
		tmp = x / fma(y, (0.16666666666666666 * (y * z_m)), z_m);
	} else {
		tmp = (x / y) * (sin(y) / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 4e-5)
		tmp = Float64(x / fma(y, Float64(0.16666666666666666 * Float64(y * z_m)), z_m));
	else
		tmp = Float64(Float64(x / y) * Float64(sin(y) / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000033e-5

   1. Initial program 98.0%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   11. Applied rewrites 76.9%

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

   if 4.00000000000000033e-5 < y

   1. Initial program 94.6%

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

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

4. Final simplification 81.0%

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

Alternative 9: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.005:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right) \cdot z\_m, y \cdot y, z\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{y \cdot z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 0.005)
    (/
     x
     (fma
      (* (fma 0.019444444444444445 (* y y) 0.16666666666666666) z_m)
      (* y y)
      z_m))
    (/ (* (sin y) x) (* y z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.005) {
		tmp = x / fma((fma(0.019444444444444445, (y * y), 0.16666666666666666) * z_m), (y * y), z_m);
	} else {
		tmp = (sin(y) * x) / (y * z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 0.005)
		tmp = Float64(x / fma(Float64(fma(0.019444444444444445, Float64(y * y), 0.16666666666666666) * z_m), Float64(y * y), z_m));
	else
		tmp = Float64(Float64(sin(y) * x) / Float64(y * z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 0.005], N[(x / N[(N[(N[(0.019444444444444445 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * z$95$m), $MachinePrecision] * N[(y * y), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.0050000000000000001

   1. Initial program 98.0%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{z + {y}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right) + z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right) \cdot {y}^{2}} + z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z, {y}^{2}, z\right)}} \]

   9. Applied rewrites 77.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, z\right)}} \]

   if 0.0050000000000000001 < y

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 86.5

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

   4. Applied rewrites 86.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.005:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right) \cdot z, y \cdot y, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 3.5e+108)
    (* (/ (fma -0.16666666666666666 (* y y) 1.0) z_m) x)
    (/ x (* (* (* y y) 0.16666666666666666) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.5e+108) {
		tmp = (fma(-0.16666666666666666, (y * y), 1.0) / z_m) * x;
	} else {
		tmp = x / (((y * y) * 0.16666666666666666) * z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 3.5e+108)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) / z_m) * x);
	else
		tmp = Float64(x / Float64(Float64(Float64(y * y) * 0.16666666666666666) * z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 3.5e+108], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(x / N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000002e108

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.0%

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

   if 3.5000000000000002e108 < y

   1. Initial program 92.9%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 84.2

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

   6. Applied rewrites 84.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 28.7

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

   9. Applied rewrites 28.7%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 28.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 61.0

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

5. Applied rewrites 61.0%

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

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024304 
(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))