Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.8%

Time: 7.0s

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: 96.0% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\frac{y}{\sin y}}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 3.4e+14)
    (* (/ (sin y) y) (/ x_m z))
    (/ (/ x_m (/ y (sin y))) z))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.4e+14) {
		tmp = (sin(y) / y) * (x_m / z);
	} else {
		tmp = (x_m / (y / sin(y))) / z;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 3.4d+14) then
        tmp = (sin(y) / y) * (x_m / z)
    else
        tmp = (x_m / (y / sin(y))) / z
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.4e+14) {
		tmp = (Math.sin(y) / y) * (x_m / z);
	} else {
		tmp = (x_m / (y / Math.sin(y))) / z;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 3.4e+14:
		tmp = (math.sin(y) / y) * (x_m / z)
	else:
		tmp = (x_m / (y / math.sin(y))) / z
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 3.4e+14)
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(y / sin(y))) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 3.4e+14)
		tmp = (sin(y) / y) * (x_m / z);
	else
		tmp = (x_m / (y / sin(y))) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 3.4e+14], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e14

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*r/ 97.5%

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

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
   4. Add Preprocessing

   if 3.4e14 < x

   1. Initial program 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 1.6e-32) (/ x_m z) (* x_m (/ (sin y) (* y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.6e-32) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (sin(y) / (y * z));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.6d-32) then
        tmp = x_m / z
    else
        tmp = x_m * (sin(y) / (y * z))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.6e-32) {
		tmp = x_m / z;
	} else {
		tmp = x_m * (Math.sin(y) / (y * z));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 1.6e-32:
		tmp = x_m / z
	else:
		tmp = x_m * (math.sin(y) / (y * z))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.6e-32)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(x_m * Float64(sin(y) / Float64(y * z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 1.6e-32)
		tmp = x_m / z;
	else
		tmp = x_m * (sin(y) / (y * z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.6e-32], N[(x$95$m / z), $MachinePrecision], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.6000000000000001e-32

   1. Initial program 95.0%

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

      1. associate-/l* 96.8%

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

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 68.9%

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

   if 1.6000000000000001e-32 < y

   1. Initial program 90.9%

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

      1. associate-*r/ 93.3%

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

      2. associate-/l/ 93.2%

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

      3. *-commutative 93.2%

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

   3. Simplified 93.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 1e-16) (/ x_m z) (* (sin y) (/ x_m (* y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1e-16) {
		tmp = x_m / z;
	} else {
		tmp = sin(y) * (x_m / (y * z));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d-16) then
        tmp = x_m / z
    else
        tmp = sin(y) * (x_m / (y * z))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1e-16) {
		tmp = x_m / z;
	} else {
		tmp = Math.sin(y) * (x_m / (y * z));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 1e-16:
		tmp = x_m / z
	else:
		tmp = math.sin(y) * (x_m / (y * z))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1e-16)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 1e-16)
		tmp = x_m / z;
	else
		tmp = sin(y) * (x_m / (y * z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1e-16], N[(x$95$m / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e-17

   1. Initial program 95.1%

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

      1. associate-/l* 96.8%

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

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 69.4%

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

   if 9.9999999999999998e-17 < y

   1. Initial program 90.6%

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

      1. associate-/l* 93.1%

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

   3. Simplified 93.1%

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

      1. associate-/l* 90.6%

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

      2. associate-*r/ 90.6%

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

      3. associate-/r* 93.1%

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

      4. associate-*l/ 93.0%

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

      5. *-commutative 93.0%

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

   6. Applied egg-rr 93.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-16}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 10^{-9}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= z 1e-9) (/ x_m (/ z t_0)) (* t_0 (/ x_m z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z <= 1e-9) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z <= 1d-9) then
        tmp = x_m / (z / t_0)
    else
        tmp = t_0 * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z <= 1e-9) {
		tmp = x_m / (z / t_0);
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if z <= 1e-9:
		tmp = x_m / (z / t_0)
	else:
		tmp = t_0 * (x_m / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z <= 1e-9)
		tmp = Float64(x_m / Float64(z / t_0));
	else
		tmp = Float64(t_0 * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z <= 1e-9)
		tmp = x_m / (z / t_0);
	else
		tmp = t_0 * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, 1e-9], N[(x$95$m / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000006e-9

   1. Initial program 91.3%

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

      1. associate-/l* 96.6%

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

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Add Preprocessing

   if 1.00000000000000006e-9 < z

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 3e-5) (* t_0 (/ x_m z)) (/ (* x_m t_0) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 3e-5) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 3d-5) then
        tmp = t_0 * (x_m / z)
    else
        tmp = (x_m * t_0) / z
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 3e-5) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 3e-5:
		tmp = t_0 * (x_m / z)
	else:
		tmp = (x_m * t_0) / z
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 3e-5)
		tmp = Float64(t_0 * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * t_0) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 3e-5)
		tmp = t_0 * (x_m / z);
	else
		tmp = (x_m * t_0) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 3e-5], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.00000000000000008e-5

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*r/ 97.5%

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

   3. Simplified 97.5%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
   4. Add Preprocessing

   if 3.00000000000000008e-5 < x

   1. Initial program 99.8%

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

4. Final simplification 98.0%

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

Alternative 6: 95.9% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* (/ (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) {
	return x_s * ((sin(y) / y) * (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 * ((sin(y) / y) * (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 * ((Math.sin(y) / y) * (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 * ((math.sin(y) / y) * (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(Float64(sin(y) / y) * 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 * ((sin(y) / y) * (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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. *-commutative 93.8%

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

   2. associate-*r/ 95.5%

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

3. Simplified 95.5%

   \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
4. Add Preprocessing

5. Final simplification 95.5%

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

Alternative 7: 61.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{x\_m}{z} + -1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5e+58) (/ x_m z) (+ 1.0 (+ (/ x_m z) -1.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5e+58) {
		tmp = x_m / z;
	} else {
		tmp = 1.0 + ((x_m / z) + -1.0);
	}
	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 (y <= 5d+58) then
        tmp = x_m / z
    else
        tmp = 1.0d0 + ((x_m / z) + (-1.0d0))
    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 (y <= 5e+58) {
		tmp = x_m / z;
	} else {
		tmp = 1.0 + ((x_m / z) + -1.0);
	}
	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 y <= 5e+58:
		tmp = x_m / z
	else:
		tmp = 1.0 + ((x_m / z) + -1.0)
	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 <= 5e+58)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(1.0 + Float64(Float64(x_m / z) + -1.0));
	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 (y <= 5e+58)
		tmp = x_m / z;
	else
		tmp = 1.0 + ((x_m / z) + -1.0);
	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[y, 5e+58], N[(x$95$m / z), $MachinePrecision], N[(1.0 + N[(N[(x$95$m / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999986e58

   1. Initial program 95.4%

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

      1. associate-/l* 97.0%

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.4%

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

   if 4.99999999999999986e58 < y

   1. Initial program 88.3%

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

      1. associate-*r/ 91.4%

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

      2. associate-/l/ 91.3%

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

      3. *-commutative 91.3%

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

   3. Simplified 91.3%

      \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 16.3%

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

      1. un-div-inv 16.3%

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

      2. clear-num 18.5%

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

   7. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 16.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{z}{x}}\right)\right)} \]

      2. expm1-udef 29.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{z}{x}}\right)} - 1} \]

      3. log1p-udef 29.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + \frac{1}{\frac{z}{x}}\right)}} - 1 \]

      4. rem-exp-log 31.9%

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\frac{z}{x}}\right)} - 1 \]

      5. clear-num 31.9%

         \[\leadsto \left(1 + \color{blue}{\frac{x}{z}}\right) - 1 \]

   9. Applied egg-rr 31.9%

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

       1. associate--l+ 31.9%

          \[\leadsto \color{blue}{1 + \left(\frac{x}{z} - 1\right)} \]

       2. +-commutative 31.9%

          \[\leadsto \color{blue}{\left(\frac{x}{z} - 1\right) + 1} \]

       3. sub-neg 31.9%

          \[\leadsto \color{blue}{\left(\frac{x}{z} + \left(-1\right)\right)} + 1 \]

       4. metadata-eval 31.9%

          \[\leadsto \left(\frac{x}{z} + \color{blue}{-1}\right) + 1 \]

   11. Applied egg-rr 31.9%

       \[\leadsto \color{blue}{\left(\frac{x}{z} + -1\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{x}{z} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{z} + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 5e+58) (/ x_m z) (+ (+ (/ x_m z) 1.0) -1.0))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 5e+58) {
		tmp = x_m / z;
	} else {
		tmp = ((x_m / z) + 1.0) + -1.0;
	}
	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 (y <= 5d+58) then
        tmp = x_m / z
    else
        tmp = ((x_m / z) + 1.0d0) + (-1.0d0)
    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 (y <= 5e+58) {
		tmp = x_m / z;
	} else {
		tmp = ((x_m / z) + 1.0) + -1.0;
	}
	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 y <= 5e+58:
		tmp = x_m / z
	else:
		tmp = ((x_m / z) + 1.0) + -1.0
	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 <= 5e+58)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) + 1.0) + -1.0);
	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 (y <= 5e+58)
		tmp = x_m / z;
	else
		tmp = ((x_m / z) + 1.0) + -1.0;
	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[y, 5e+58], N[(x$95$m / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999986e58

   1. Initial program 95.4%

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

      1. associate-/l* 97.0%

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.4%

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

   if 4.99999999999999986e58 < y

   1. Initial program 88.3%

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

      1. associate-*r/ 91.4%

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

      2. associate-/l/ 91.3%

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

      3. *-commutative 91.3%

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

   3. Simplified 91.3%

      \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 16.3%

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

      1. un-div-inv 16.3%

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

      2. clear-num 18.5%

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

   7. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 16.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{z}{x}}\right)\right)} \]

      2. expm1-udef 29.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{z}{x}}\right)} - 1} \]

      3. log1p-udef 29.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + \frac{1}{\frac{z}{x}}\right)}} - 1 \]

      4. rem-exp-log 31.9%

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\frac{z}{x}}\right)} - 1 \]

      5. clear-num 31.9%

         \[\leadsto \left(1 + \color{blue}{\frac{x}{z}}\right) - 1 \]

   9. Applied egg-rr 31.9%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{z}\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 21.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ 1.0 (/ 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) {
	return x_s * (1.0 / (z / x_m));
}

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 * (1.0d0 / (z / x_m))
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 * (1.0 / (z / x_m));
}

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(1.0 / Float64(z / x_m)))
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 * (1.0 / (z / x_m));
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[(1.0 / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-*r/ 95.8%

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

   2. associate-/l/ 90.3%

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

   3. *-commutative 90.3%

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

3. Simplified 90.3%

   \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y \cdot z}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 55.7%

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

   1. un-div-inv 55.8%

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

   2. clear-num 56.2%

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

7. Applied egg-rr 56.2%

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

8. Final simplification 56.2%

   \[\leadsto \frac{1}{\frac{z}{x}} \]
9. Add Preprocessing

Alternative 10: 57.8% accurate, 35.7× 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 1 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 93.8%

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

   1. associate-/l* 95.8%

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

3. Simplified 95.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 55.8%

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

6. Final simplification 55.8%

   \[\leadsto \frac{x}{z} \]
7. Add Preprocessing

Developer target: 99.6% accurate, 0.9× 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 2024041 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))