Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.8%

Time: 8.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% 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: 99.8% accurate, 0.9× speedup

Math

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

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 1.0) (/ (/ x_m z) (/ y (sin y))) (/ (* x_m (/ (sin y) y)) z))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 97.1%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if 1 < x

   1. Initial program 99.7%

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

Alternative 2: 40.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -2e-302)
      (/ (* x_m (* (* y y) -0.16666666666666666)) z)
      (if (<= t_0 5e-306) (/ (* y x_m) (* z y)) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = (x_m * ((y * y) * -0.16666666666666666)) / z;
	} else if (t_0 <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-2d-302)) then
        tmp = (x_m * ((y * y) * (-0.16666666666666666d0))) / z
    else if (t_0 <= 5d-306) then
        tmp = (y * x_m) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = (x_m * ((y * y) * -0.16666666666666666)) / z;
	} else if (t_0 <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -2e-302:
		tmp = (x_m * ((y * y) * -0.16666666666666666)) / z
	elif t_0 <= 5e-306:
		tmp = (y * x_m) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -2e-302)
		tmp = Float64(Float64(x_m * Float64(Float64(y * y) * -0.16666666666666666)) / z);
	elseif (t_0 <= 5e-306)
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -2e-302)
		tmp = (x_m * ((y * y) * -0.16666666666666666)) / z;
	elseif (t_0 <= 5e-306)
		tmp = (y * x_m) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-302], N[(N[(x$95$m * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-306], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.9999999999999999e-302

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 5.1%

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

   if -1.9999999999999999e-302 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.99999999999999998e-306

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.8

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

   7. Applied rewrites 70.8%

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

   if 4.99999999999999998e-306 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 3: 40.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(y \cdot y\right)}{z} \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -2e-302)
      (* (/ (* -0.16666666666666666 (* y y)) z) x_m)
      (if (<= t_0 5e-306) (/ (* y x_m) (* z y)) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = ((-0.16666666666666666 * (y * y)) / z) * x_m;
	} else if (t_0 <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-2d-302)) then
        tmp = (((-0.16666666666666666d0) * (y * y)) / z) * x_m
    else if (t_0 <= 5d-306) then
        tmp = (y * x_m) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = ((-0.16666666666666666 * (y * y)) / z) * x_m;
	} else if (t_0 <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -2e-302:
		tmp = ((-0.16666666666666666 * (y * y)) / z) * x_m
	elif t_0 <= 5e-306:
		tmp = (y * x_m) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -2e-302)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(y * y)) / z) * x_m);
	elseif (t_0 <= 5e-306)
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -2e-302)
		tmp = ((-0.16666666666666666 * (y * y)) / z) * x_m;
	elseif (t_0 <= 5e-306)
		tmp = (y * x_m) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-302], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-306], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.9999999999999999e-302

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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 60.7

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

   5. Applied rewrites 60.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}}{z} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.5

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 5.1%

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

   if -1.9999999999999999e-302 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.99999999999999998e-306

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.8

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

   7. Applied rewrites 70.8%

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

   if 4.99999999999999998e-306 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 4: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999:\\ \;\;\;\;\frac{\sin y}{\frac{z}{x\_m} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot z}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.9999999)
    (/ (sin y) (* (/ z x_m) y))
    (/ x_m (* (fma 0.16666666666666666 (* y y) 1.0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9999999) {
		tmp = sin(y) / ((z / x_m) * y);
	} else {
		tmp = x_m / (fma(0.16666666666666666, (y * y), 1.0) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999)
		tmp = Float64(sin(y) / Float64(Float64(z / x_m) * y));
	else
		tmp = Float64(x_m / Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999], N[(N[Sin[y], $MachinePrecision] / N[(N[(z / x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999999900000000053

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. 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 93.8

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

   4. Applied rewrites 93.8%

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

   if 0.999999900000000053 < (/.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. 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. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right)} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 5: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot z}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.9999999)
    (/ (* (sin y) x_m) (* z y))
    (/ x_m (* (fma 0.16666666666666666 (* y y) 1.0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9999999) {
		tmp = (sin(y) * x_m) / (z * y);
	} else {
		tmp = x_m / (fma(0.16666666666666666, (y * y), 1.0) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999)
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999999900000000053

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 92.0

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

   4. Applied rewrites 92.0%

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

   if 0.999999900000000053 < (/.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. 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. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right)} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 6: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= (/ (* x_m (/ (sin y) y)) z) 5e-306)
    (/ (* 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 (((x_m * (sin(y) / y)) / z) <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 5d-306) then
        tmp = (y * x_m) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 5e-306) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 5e-306:
		tmp = (y * x_m) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 5e-306)
		tmp = Float64(Float64(y * 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 (((x_m * (sin(y) / y)) / z) <= 5e-306)
		tmp = (y * x_m) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-306], N[(N[(y * x$95$m), $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) < 4.99999999999999998e-306

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.9

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

   7. Applied rewrites 57.9%

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

   if 4.99999999999999998e-306 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 7: 96.2% accurate, 1.0× speedup

Math

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

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 (/ (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) {
	return x_s * ((x_m * (sin(y) / y)) / 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 * (sin(y) / y)) / 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 * (Math.sin(y) / y)) / 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 * (math.sin(y) / y)) / z)

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

Alternative 8: 58.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 2.5e+67)
    (* (fma (* y y) -0.16666666666666666 1.0) (/ x_m z))
    (/ (* y x_m) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2.5e+67) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x_m / z);
	} else {
		tmp = (y * x_m) / (z * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 2.5e+67)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 2.5e+67], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999988e67

   1. Initial program 98.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 65.6

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

   5. Applied rewrites 65.6%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}}{z} \]
   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. lift-/.f64 N/A

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

      6. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if 2.49999999999999988e67 < 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 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.6

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

   7. Applied rewrites 29.6%

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

Alternative 9: 57.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.5e+43)
    (* (/ (fma -0.16666666666666666 (* y y) 1.0) z) x_m)
    (/ (* y x_m) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.5e+43) {
		tmp = (fma(-0.16666666666666666, (y * y), 1.0) / z) * x_m;
	} else {
		tmp = (y * x_m) / (z * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.5e+43)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) / z) * x_m);
	else
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.5e+43], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000001e43

   1. Initial program 98.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 64.4%

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

   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 66.3%

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

   if 3.5000000000000001e43 < y

   1. Initial program 94.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 28.1

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

   7. Applied rewrites 28.1%

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

Alternative 10: 66.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{x\_m}{z}}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \end{array} \]

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * ((x_m / z) / fma(0.16666666666666666, (y * y), 1.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) / fma(0.16666666666666666, Float64(y * y), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 67.5

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

7. Applied rewrites 67.5%

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

Alternative 11: 66.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot z} \end{array} \]

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 (* (fma 0.16666666666666666 (* y y) 1.0) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / (fma(0.16666666666666666, (y * y), 1.0) * 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 / Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * 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 / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{\frac{x}{z}}{\color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}, {y}^{2}, 1\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\frac{7}{360} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7}{360}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7}{360}, y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right)} \]

   9. lower-*.f64 67.1

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right)} \]

7. Applied rewrites 67.1%

   \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 67.5%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/l/ N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 67.2

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

12. Applied rewrites 67.2%

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

Alternative 12: 58.8% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 56.6

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

5. Applied rewrites 56.6%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024317 
(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))