Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.8%

Time: 6.9s

Alternatives: 7

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, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ \end{array} \end{array} \]

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 8e-6) (/ x_m (* z (/ y (sin y)))) (/ (* (/ (sin y) 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 <= 8e-6) {
		tmp = x_m / (z * (y / sin(y)));
	} else {
		tmp = ((sin(y) / y) * 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 <= 8d-6) then
        tmp = x_m / (z * (y / sin(y)))
    else
        tmp = ((sin(y) / y) * 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 <= 8e-6) {
		tmp = x_m / (z * (y / Math.sin(y)));
	} else {
		tmp = ((Math.sin(y) / y) * 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 <= 8e-6:
		tmp = x_m / (z * (y / math.sin(y)))
	else:
		tmp = ((math.sin(y) / y) * 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 (x_m <= 8e-6)
		tmp = Float64(x_m / Float64(z * Float64(y / sin(y))));
	else
		tmp = Float64(Float64(Float64(sin(y) / y) * 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 <= 8e-6)
		tmp = x_m / (z * (y / sin(y)));
	else
		tmp = ((sin(y) / y) * 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[x$95$m, 8e-6], N[(x$95$m / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999964e-6

   1. Initial program 93.8%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if 7.99999999999999964e-6 < x

   1. Initial program 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-78}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\sin y}{y}\\ \end{array} \end{array} \]

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 (<= z 3e-78) (/ x_m (* z (/ y (sin y)))) (* (/ x_m z) (/ (sin y) 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 (z <= 3e-78) {
		tmp = x_m / (z * (y / sin(y)));
	} else {
		tmp = (x_m / z) * (sin(y) / y);
	}
	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 (z <= 3d-78) then
        tmp = x_m / (z * (y / sin(y)))
    else
        tmp = (x_m / z) * (sin(y) / y)
    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 (z <= 3e-78) {
		tmp = x_m / (z * (y / Math.sin(y)));
	} else {
		tmp = (x_m / z) * (Math.sin(y) / y);
	}
	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 z <= 3e-78:
		tmp = x_m / (z * (y / math.sin(y)))
	else:
		tmp = (x_m / z) * (math.sin(y) / 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 (z <= 3e-78)
		tmp = Float64(x_m / Float64(z * Float64(y / sin(y))));
	else
		tmp = Float64(Float64(x_m / z) * Float64(sin(y) / y));
	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 (z <= 3e-78)
		tmp = x_m / (z * (y / sin(y)));
	else
		tmp = (x_m / z) * (sin(y) / y);
	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[z, 3e-78], N[(x$95$m / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.99999999999999988e-78

   1. Initial program 93.2%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   if 2.99999999999999988e-78 < z

   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. *-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. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.00022:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \end{array} \end{array} \]

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 0.00022)
    (/ x_m (fma (* (* y y) z) 0.16666666666666666 z))
    (* (/ x_m (* z y)) (sin 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 <= 0.00022) {
		tmp = x_m / fma(((y * y) * z), 0.16666666666666666, z);
	} else {
		tmp = (x_m / (z * y)) * sin(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 <= 0.00022)
		tmp = Float64(x_m / fma(Float64(Float64(y * y) * z), 0.16666666666666666, z));
	else
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(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, 0.00022], N[(x$95$m / N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * 0.16666666666666666 + z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000008e-4

   1. Initial program 97.0%

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

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

   if 2.20000000000000008e-4 < y

   1. Initial program 90.3%

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

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.4

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

   6. Applied rewrites 86.4%

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

Alternative 4: 95.8% accurate, 1.0× speedup

Math

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

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) (/ (sin y) y))))

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) * (sin(y) / y));
}

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) * (sin(y) / y))
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) * (Math.sin(y) / y));
}

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) * (math.sin(y) / y))

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) * Float64(sin(y) / y)))
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) * (sin(y) / y));
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[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. lift-/.f64 N/A

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

   2. 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. lower-*.f64 N/A

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

   6. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 5: 57.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{z \cdot x\_m}\\ \end{array} \end{array} \]

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 3.2e+64)
    (* (fma (* -0.16666666666666666 y) y 1.0) (/ x_m z))
    (/ (* x_m x_m) (* z x_m)))))

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 <= 3.2e+64) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * (x_m / z);
	} else {
		tmp = (x_m * x_m) / (z * x_m);
	}
	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 <= 3.2e+64)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(z * x_m));
	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, 3.2e+64], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.20000000000000019e64

   1. Initial program 96.9%

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

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

      6. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.6

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 72.6%

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

   if 3.20000000000000019e64 < y

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 7.8

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

   5. Applied rewrites 7.8%

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

   7. Applied rewrites 7.8%

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

   9. Applied rewrites 11.2%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 4.6× speedup

Math

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

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 (fma (* (* y y) z) 0.16666666666666666 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(((y * y) * z), 0.16666666666666666, 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 / fma(Float64(Float64(y * y) * z), 0.16666666666666666, 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[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * 0.16666666666666666 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. lift-/.f64 N/A

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

   2. 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. associate-/r/ N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 94.2

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 68.2

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

7. Applied rewrites 68.2%

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

Alternative 7: 58.5% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 62.3

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

5. Applied rewrites 62.3%

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

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024297 
(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))