Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.6%

Time: 8.2s

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: 96.0% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-76}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 -1e-55)
     (/ x (/ z t_0))
     (if (<= t_1 2e-76) (* t_0 (/ x z)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -1e-55) {
		tmp = x / (z / t_0);
	} else if (t_1 <= 2e-76) {
		tmp = t_0 * (x / z);
	} 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 = sin(y) / y
    t_1 = (x * t_0) / z
    if (t_1 <= (-1d-55)) then
        tmp = x / (z / t_0)
    else if (t_1 <= 2d-76) then
        tmp = t_0 * (x / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -1e-55) {
		tmp = x / (z / t_0);
	} else if (t_1 <= 2e-76) {
		tmp = t_0 * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = (x * t_0) / z
	tmp = 0
	if t_1 <= -1e-55:
		tmp = x / (z / t_0)
	elif t_1 <= 2e-76:
		tmp = t_0 * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= -1e-55)
		tmp = Float64(x / Float64(z / t_0));
	elseif (t_1 <= 2e-76)
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	t_1 = (x * t_0) / z;
	tmp = 0.0;
	if (t_1 <= -1e-55)
		tmp = x / (z / t_0);
	elseif (t_1 <= 2e-76)
		tmp = t_0 * (x / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-55], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-76], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.99999999999999995e-56

   1. Initial program 99.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   if -9.99999999999999995e-56 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.99999999999999985e-76

   1. Initial program 93.5%

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

      1. associate-*l/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 1.99999999999999985e-76 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-31} \lor \neg \left(z \leq 1.6 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3e-31) (not (<= z 1.6e-53)))
   (* (/ (sin y) y) (/ x z))
   (* x (/ (/ (sin y) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-31) || !(z <= 1.6e-53)) {
		tmp = (sin(y) / y) * (x / z);
	} else {
		tmp = x * ((sin(y) / z) / y);
	}
	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) :: tmp
    if ((z <= (-3d-31)) .or. (.not. (z <= 1.6d-53))) then
        tmp = (sin(y) / y) * (x / z)
    else
        tmp = x * ((sin(y) / z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-31) || !(z <= 1.6e-53)) {
		tmp = (Math.sin(y) / y) * (x / z);
	} else {
		tmp = x * ((Math.sin(y) / z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3e-31) or not (z <= 1.6e-53):
		tmp = (math.sin(y) / y) * (x / z)
	else:
		tmp = x * ((math.sin(y) / z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3e-31) || !(z <= 1.6e-53))
		tmp = Float64(Float64(sin(y) / y) * Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(sin(y) / z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3e-31) || ~((z <= 1.6e-53)))
		tmp = (sin(y) / y) * (x / z);
	else
		tmp = x * ((sin(y) / z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3e-31], N[Not[LessEqual[z, 1.6e-53]], $MachinePrecision]], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999981e-31 or 1.6e-53 < z

   1. Initial program 99.0%

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

      1. associate-*l/ 99.8%

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

      2. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   if -2.99999999999999981e-31 < z < 1.6e-53

   1. Initial program 92.3%

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

      1. associate-*r/ 99.6%

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

      2. associate-/l/ 82.3%

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

      3. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-31} \lor \neg \left(z \leq 1.6 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+83} \lor \neg \left(z \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= z -2e+83) (not (<= z 5e-35))) (* t_0 (/ x z)) (/ x (/ z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((z <= -2e+83) || !(z <= 5e-35)) {
		tmp = t_0 * (x / z);
	} else {
		tmp = x / (z / t_0);
	}
	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) :: tmp
    t_0 = sin(y) / y
    if ((z <= (-2d+83)) .or. (.not. (z <= 5d-35))) then
        tmp = t_0 * (x / z)
    else
        tmp = x / (z / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if ((z <= -2e+83) || !(z <= 5e-35)) {
		tmp = t_0 * (x / z);
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if (z <= -2e+83) or not (z <= 5e-35):
		tmp = t_0 * (x / z)
	else:
		tmp = x / (z / t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((z <= -2e+83) || !(z <= 5e-35))
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = Float64(x / Float64(z / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if ((z <= -2e+83) || ~((z <= 5e-35)))
		tmp = t_0 * (x / z);
	else
		tmp = x / (z / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[z, -2e+83], N[Not[LessEqual[z, 5e-35]], $MachinePrecision]], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000006e83 or 4.99999999999999964e-35 < z

   1. Initial program 99.4%

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

      1. associate-*l/ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -2.00000000000000006e83 < z < 4.99999999999999964e-35

   1. Initial program 93.6%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+83} \lor \neg \left(z \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

   1. associate-*r/ 95.3%

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

3. Simplified 95.3%

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

4. Final simplification 95.3%

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

Alternative 5: 61.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} \cdot \frac{y \cdot z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.9e-12) (/ x z) (/ 1.0 (* (/ 1.0 y) (/ (* y z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.9e-12) {
		tmp = x / z;
	} else {
		tmp = 1.0 / ((1.0 / y) * ((y * z) / x));
	}
	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) :: tmp
    if (y <= 3.9d-12) then
        tmp = x / z
    else
        tmp = 1.0d0 / ((1.0d0 / y) * ((y * z) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.9e-12) {
		tmp = x / z;
	} else {
		tmp = 1.0 / ((1.0 / y) * ((y * z) / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.9e-12:
		tmp = x / z
	else:
		tmp = 1.0 / ((1.0 / y) * ((y * z) / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.9e-12)
		tmp = Float64(x / z);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / y) * Float64(Float64(y * z) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.9e-12)
		tmp = x / z;
	else
		tmp = 1.0 / ((1.0 / y) * ((y * z) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.9e-12], N[(x / z), $MachinePrecision], N[(1.0 / N[(N[(1.0 / y), $MachinePrecision] * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.89999999999999994e-12

   1. Initial program 97.7%

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

   2. Taylor expanded in y around 0 65.7%

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

   if 3.89999999999999994e-12 < y

   1. Initial program 92.2%

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

      1. associate-*r/ 94.0%

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

      2. associate-/l/ 93.5%

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

      3. associate-/r* 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in y around 0 21.8%

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

      1. associate-*r/ 21.7%

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

      2. clear-num 22.7%

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

   6. Applied egg-rr 22.7%

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

      1. associate-*r/ 22.8%

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

      2. *-commutative 22.8%

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

      3. associate-/l* 27.3%

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

      4. *-un-lft-identity 27.3%

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

      5. times-frac 37.4%

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

      6. *-commutative 37.4%

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

   8. Applied egg-rr 37.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{y} \cdot \frac{z \cdot y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} \cdot \frac{y \cdot z}{x}}\\ \end{array} \]

Alternative 6: 61.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+22) (/ x z) (/ y (/ (* y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+22) {
		tmp = x / z;
	} else {
		tmp = y / ((y * z) / x);
	}
	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) :: tmp
    if (y <= 2d+22) then
        tmp = x / z
    else
        tmp = y / ((y * z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+22) {
		tmp = x / z;
	} else {
		tmp = y / ((y * z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+22:
		tmp = x / z
	else:
		tmp = y / ((y * z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+22)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(Float64(y * z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+22)
		tmp = x / z;
	else
		tmp = y / ((y * z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+22], N[(x / z), $MachinePrecision], N[(y / N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e22

   1. Initial program 97.7%

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

   2. Taylor expanded in y around 0 64.7%

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

   if 2e22 < y

   1. Initial program 91.1%

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

      1. associate-*r/ 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in y around 0 19.2%

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

      1. *-un-lft-identity 19.2%

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

      2. *-inverses 19.2%

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

      3. un-div-inv 19.2%

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

      4. times-frac 25.2%

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

      5. associate-/l* 36.8%

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

      6. *-commutative 36.8%

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

   6. Applied egg-rr 36.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \end{array} \]

Alternative 7: 57.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / z
end function

Java

public static double code(double x, double y, double z) {
	return x / z;
}

Python

def code(x, y, z):
	return x / z

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in y around 0 55.3%

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

3. Final simplification 55.3%

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

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 2023305 
(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))