Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.3%

Time: 6.9s

Alternatives: 10

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_0}{z\_m}\\ \end{array} \end{array} \end{array} \]

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)))
   (* z_s (if (<= z_m 3.2e-91) (* (/ t_0 z_m) x) (/ (* x t_0) 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;
	double tmp;
	if (z_m <= 3.2e-91) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (x * t_0) / 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
    if (z_m <= 3.2d-91) then
        tmp = (t_0 / z_m) * x
    else
        tmp = (x * t_0) / 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;
	double tmp;
	if (z_m <= 3.2e-91) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (x * t_0) / 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
	tmp = 0
	if z_m <= 3.2e-91:
		tmp = (t_0 / z_m) * x
	else:
		tmp = (x * t_0) / 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(sin(y) / y)
	tmp = 0.0
	if (z_m <= 3.2e-91)
		tmp = Float64(Float64(t_0 / z_m) * x);
	else
		tmp = Float64(Float64(x * t_0) / 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;
	tmp = 0.0;
	if (z_m <= 3.2e-91)
		tmp = (t_0 / z_m) * x;
	else
		tmp = (x * t_0) / 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[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 3.2e-91], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999996e-91

   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 3.19999999999999996e-91 < z

   1. Initial program 99.8%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot t\_0 \leq -1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{x}{y}}{z\_m} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

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)))
   (*
    z_s
    (if (<= (* x t_0) -1e-54) (* (/ (/ x y) z_m) (sin y)) (* (/ t_0 z_m) x)))))

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;
	double tmp;
	if ((x * t_0) <= -1e-54) {
		tmp = ((x / y) / z_m) * sin(y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	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
    if ((x * t_0) <= (-1d-54)) then
        tmp = ((x / y) / z_m) * sin(y)
    else
        tmp = (t_0 / z_m) * x
    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;
	double tmp;
	if ((x * t_0) <= -1e-54) {
		tmp = ((x / y) / z_m) * Math.sin(y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	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
	tmp = 0
	if (x * t_0) <= -1e-54:
		tmp = ((x / y) / z_m) * math.sin(y)
	else:
		tmp = (t_0 / z_m) * x
	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(sin(y) / y)
	tmp = 0.0
	if (Float64(x * t_0) <= -1e-54)
		tmp = Float64(Float64(Float64(x / y) / z_m) * sin(y));
	else
		tmp = Float64(Float64(t_0 / z_m) * x);
	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;
	tmp = 0.0;
	if ((x * t_0) <= -1e-54)
		tmp = ((x / y) / z_m) * sin(y);
	else
		tmp = (t_0 / z_m) * x;
	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[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[N[(x * t$95$0), $MachinePrecision], -1e-54], N[(N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -1e-54

   1. Initial program 99.8%

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 73.2

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

   4. Applied rewrites 73.2%

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

   if -1e-54 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 95.3%

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

Alternative 3: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+229}:\\ \;\;\;\;\frac{\frac{x}{y}}{z\_m} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 (<= y 2.6e-16)
    (/ x z_m)
    (if (<= y 3.2e+229)
      (* (/ (/ x y) z_m) (sin y))
      (/ (* (sin y) x) (* z_m 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 (y <= 2.6e-16) {
		tmp = x / z_m;
	} else if (y <= 3.2e+229) {
		tmp = ((x / y) / z_m) * sin(y);
	} else {
		tmp = (sin(y) * x) / (z_m * 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 (y <= 2.6d-16) then
        tmp = x / z_m
    else if (y <= 3.2d+229) then
        tmp = ((x / y) / z_m) * sin(y)
    else
        tmp = (sin(y) * x) / (z_m * 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 (y <= 2.6e-16) {
		tmp = x / z_m;
	} else if (y <= 3.2e+229) {
		tmp = ((x / y) / z_m) * Math.sin(y);
	} else {
		tmp = (Math.sin(y) * x) / (z_m * 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 y <= 2.6e-16:
		tmp = x / z_m
	elif y <= 3.2e+229:
		tmp = ((x / y) / z_m) * math.sin(y)
	else:
		tmp = (math.sin(y) * x) / (z_m * 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 (y <= 2.6e-16)
		tmp = Float64(x / z_m);
	elseif (y <= 3.2e+229)
		tmp = Float64(Float64(Float64(x / y) / z_m) * sin(y));
	else
		tmp = Float64(Float64(sin(y) * x) / Float64(z_m * 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 (y <= 2.6e-16)
		tmp = x / z_m;
	elseif (y <= 3.2e+229)
		tmp = ((x / y) / z_m) * sin(y);
	else
		tmp = (sin(y) * x) / (z_m * 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[y, 2.6e-16], N[(x / z$95$m), $MachinePrecision], If[LessEqual[y, 3.2e+229], N[(N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.5999999999999998e-16

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 2.5999999999999998e-16 < y < 3.1999999999999998e229

   1. Initial program 95.6%

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if 3.1999999999999998e229 < y

   1. Initial program 74.8%

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

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 4: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+229}:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 (<= y 2.6e-16)
    (/ x z_m)
    (if (<= y 3.2e+229)
      (* (/ (sin y) z_m) (/ x y))
      (/ (* (sin y) x) (* z_m 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 (y <= 2.6e-16) {
		tmp = x / z_m;
	} else if (y <= 3.2e+229) {
		tmp = (sin(y) / z_m) * (x / y);
	} else {
		tmp = (sin(y) * x) / (z_m * 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 (y <= 2.6d-16) then
        tmp = x / z_m
    else if (y <= 3.2d+229) then
        tmp = (sin(y) / z_m) * (x / y)
    else
        tmp = (sin(y) * x) / (z_m * 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 (y <= 2.6e-16) {
		tmp = x / z_m;
	} else if (y <= 3.2e+229) {
		tmp = (Math.sin(y) / z_m) * (x / y);
	} else {
		tmp = (Math.sin(y) * x) / (z_m * 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 y <= 2.6e-16:
		tmp = x / z_m
	elif y <= 3.2e+229:
		tmp = (math.sin(y) / z_m) * (x / y)
	else:
		tmp = (math.sin(y) * x) / (z_m * 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 (y <= 2.6e-16)
		tmp = Float64(x / z_m);
	elseif (y <= 3.2e+229)
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x / y));
	else
		tmp = Float64(Float64(sin(y) * x) / Float64(z_m * 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 (y <= 2.6e-16)
		tmp = x / z_m;
	elseif (y <= 3.2e+229)
		tmp = (sin(y) / z_m) * (x / y);
	else
		tmp = (sin(y) * x) / (z_m * 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[y, 2.6e-16], N[(x / z$95$m), $MachinePrecision], If[LessEqual[y, 3.2e+229], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.5999999999999998e-16

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 2.5999999999999998e-16 < y < 3.1999999999999998e229

   1. Initial program 95.6%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 95.7

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

   6. Applied rewrites 95.7%

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

   if 3.1999999999999998e229 < y

   1. Initial program 74.8%

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

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 5: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 (<= y 1.7e-13) (/ x z_m) (/ (* (sin y) x) (* z_m 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 (y <= 1.7e-13) {
		tmp = x / z_m;
	} else {
		tmp = (sin(y) * x) / (z_m * 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 (y <= 1.7d-13) then
        tmp = x / z_m
    else
        tmp = (sin(y) * x) / (z_m * 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 (y <= 1.7e-13) {
		tmp = x / z_m;
	} else {
		tmp = (Math.sin(y) * x) / (z_m * 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 y <= 1.7e-13:
		tmp = x / z_m
	else:
		tmp = (math.sin(y) * x) / (z_m * 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 (y <= 1.7e-13)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(sin(y) * x) / Float64(z_m * 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 (y <= 1.7e-13)
		tmp = x / z_m;
	else
		tmp = (sin(y) * x) / (z_m * 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[y, 1.7e-13], N[(x / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000008e-13

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 1.70000000000000008e-13 < y

   1. Initial program 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

Alternative 6: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.00033:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot x, y \cdot y, x\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z\_m \cdot y} \cdot x\\ \end{array} \end{array} \]

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 0.00033)
    (/ (fma (* -0.16666666666666666 x) (* y y) x) z_m)
    (* (/ (sin y) (* z_m y)) x))))

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 <= 0.00033) {
		tmp = fma((-0.16666666666666666 * x), (y * y), x) / z_m;
	} else {
		tmp = (sin(y) / (z_m * y)) * x;
	}
	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 <= 0.00033)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), Float64(y * y), x) / z_m);
	else
		tmp = Float64(Float64(sin(y) / Float64(z_m * y)) * x);
	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, 0.00033], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e-4

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.0%

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

   if 3.3e-4 < y

   1. Initial program 90.7%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.9

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

   4. Applied rewrites 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. remove-double-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-lft-neg-out N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.9

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

   6. Applied rewrites 89.9%

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

Alternative 7: 59.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot x, y \cdot y, x\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z\_m}\\ \end{array} \end{array} \]

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 2e+33)
    (/ (fma (* -0.16666666666666666 x) (* y y) x) z_m)
    (* (- 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 <= 2e+33) {
		tmp = fma((-0.16666666666666666 * x), (y * y), x) / z_m;
	} else {
		tmp = -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 <= 2e+33)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), Float64(y * y), x) / z_m);
	else
		tmp = Float64(Float64(-y) * Float64(x / Float64(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, 2e+33], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] / z$95$m), $MachinePrecision], N[((-y) * N[(x / N[((-y) * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e33

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.4%

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

   if 1.9999999999999999e33 < y

   1. Initial program 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.9

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

   7. Applied rewrites 27.9%

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

Alternative 8: 59.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z\_m}\\ \end{array} \end{array} \]

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 2e+33)
    (* (/ (fma -0.16666666666666666 (* y y) 1.0) z_m) x)
    (* (- 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 <= 2e+33) {
		tmp = (fma(-0.16666666666666666, (y * y), 1.0) / z_m) * x;
	} else {
		tmp = -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 <= 2e+33)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) / z_m) * x);
	else
		tmp = Float64(Float64(-y) * Float64(x / Float64(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, 2e+33], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[((-y) * N[(x / N[((-y) * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e33

   1. Initial program 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   if 1.9999999999999999e33 < y

   1. Initial program 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.9

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

   7. Applied rewrites 27.9%

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

Alternative 9: 62.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z\_m}\\ \end{array} \end{array} \]

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 5e+52) (/ x z_m) (* (- 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 <= 5e+52) {
		tmp = x / z_m;
	} else {
		tmp = -y * (x / (-y * 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 (y <= 5d+52) then
        tmp = x / z_m
    else
        tmp = -y * (x / (-y * 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 (y <= 5e+52) {
		tmp = x / z_m;
	} else {
		tmp = -y * (x / (-y * 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 y <= 5e+52:
		tmp = x / z_m
	else:
		tmp = -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 <= 5e+52)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(-y) * Float64(x / Float64(Float64(-y) * 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 (y <= 5e+52)
		tmp = x / z_m;
	else
		tmp = -y * (x / (-y * 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[y, 5e+52], N[(x / z$95$m), $MachinePrecision], N[((-y) * N[(x / N[((-y) * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5e52

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 5e52 < y

   1. Initial program 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.8

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

   7. Applied rewrites 28.8%

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

Alternative 10: 58.3% 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 96.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 60.3

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

5. Applied rewrites 60.3%

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

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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))