Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 97.8%

Time: 8.6s

Alternatives: 12

Speedup: 0.5×

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: 96.1% 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: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= (/ (* x t_0) z) 4e+39) (* 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 (((x * t_0) / z) <= 4e+39) {
		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 (((x * t_0) / z) <= 4d+39) 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 (((x * t_0) / z) <= 4e+39) {
		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 ((x * t_0) / z) <= 4e+39:
		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 (Float64(Float64(x * t_0) / z) <= 4e+39)
		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 (((x * t_0) / z) <= 4e+39)
		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[LessEqual[N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], 4e+39], 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 (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 3.99999999999999976e39

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   if 3.99999999999999976e39 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   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%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 2e-296) (* (sin y) (/ (/ x y) z)) (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= 2e-296) {
		tmp = sin(y) * ((x / y) / z);
	} else {
		tmp = t_0 * (x / z);
	}
	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 (t_0 <= 2d-296) then
        tmp = sin(y) * ((x / y) / z)
    else
        tmp = t_0 * (x / z)
    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 (t_0 <= 2e-296) {
		tmp = Math.sin(y) * ((x / y) / z);
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if t_0 <= 2e-296:
		tmp = math.sin(y) * ((x / y) / z)
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= 2e-296)
		tmp = Float64(sin(y) * Float64(Float64(x / y) / z));
	else
		tmp = Float64(t_0 * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (t_0 <= 2e-296)
		tmp = sin(y) * ((x / y) / z);
	else
		tmp = t_0 * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-296], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2e-296

   1. Initial program 98.9%

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

      1. associate-/l* 86.2%

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

   3. Simplified 86.2%

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

      1. clear-num 86.1%

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

      2. associate-/r/ 86.1%

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

      3. clear-num 86.1%

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

   5. Applied egg-rr 86.1%

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

   6. Taylor expanded in x around 0 86.1%

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

      1. associate-*r/ 86.1%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

   if 2e-296 < (/.f64 (sin.f64 y) y)

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*r/ 99.0%

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

   3. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (sin y) y) -2e-289)
   (* (sin y) (/ (/ x y) z))
   (/ x (* z (/ y (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= -2e-289) {
		tmp = sin(y) * ((x / y) / z);
	} else {
		tmp = x / (z * (y / sin(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 ((sin(y) / y) <= (-2d-289)) then
        tmp = sin(y) * ((x / y) / z)
    else
        tmp = x / (z * (y / sin(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= -2e-289) {
		tmp = Math.sin(y) * ((x / y) / z);
	} else {
		tmp = x / (z * (y / Math.sin(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.sin(y) / y) <= -2e-289:
		tmp = math.sin(y) * ((x / y) / z)
	else:
		tmp = x / (z * (y / math.sin(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= -2e-289)
		tmp = Float64(sin(y) * Float64(Float64(x / y) / z));
	else
		tmp = Float64(x / Float64(z * Float64(y / sin(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= -2e-289)
		tmp = sin(y) * ((x / y) / z);
	else
		tmp = x / (z * (y / sin(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], -2e-289], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -2e-289

   1. Initial program 99.4%

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

      1. associate-/l* 84.0%

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

   3. Simplified 84.0%

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

      1. clear-num 83.9%

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

      2. associate-/r/ 83.9%

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

      3. clear-num 83.9%

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

   5. Applied egg-rr 83.9%

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

   6. Taylor expanded in x around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. associate-/r* 99.4%

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

   8. Simplified 99.4%

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

   if -2e-289 < (/.f64 (sin.f64 y) y)

   1. Initial program 97.0%

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

      1. associate-/l* 99.0%

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

   3. Simplified 99.0%

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

      1. clear-num 98.8%

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

      2. associate-/r/ 99.0%

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

      3. clear-num 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-289}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternative 4: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.85e-8) (/ x z) (* (sin y) (/ (/ x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e-8) {
		tmp = x / z;
	} else {
		tmp = sin(y) * ((x / y) / z);
	}
	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 <= 1.85d-8) then
        tmp = x / z
    else
        tmp = sin(y) * ((x / y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e-8) {
		tmp = x / z;
	} else {
		tmp = Math.sin(y) * ((x / y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.85e-8:
		tmp = x / z
	else:
		tmp = math.sin(y) * ((x / y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.85e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(sin(y) * Float64(Float64(x / y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.85e-8)
		tmp = x / z;
	else
		tmp = sin(y) * ((x / y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.85e-8], N[(x / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.85e-8

   1. Initial program 98.2%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 86.5%

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

      3. associate-/l/ 80.9%

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

      4. associate-/r/ 77.3%

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

      5. associate-/r* 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around 0 73.3%

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

   if 1.85e-8 < y

   1. Initial program 94.8%

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

      1. associate-/l* 91.2%

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

   3. Simplified 91.2%

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

      1. clear-num 91.0%

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

      2. associate-/r/ 91.1%

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

      3. clear-num 91.0%

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

   5. Applied egg-rr 91.0%

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

   6. Taylor expanded in x around 0 91.0%

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

      1. associate-*r/ 91.1%

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

      2. associate-/r* 95.0%

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

   8. Simplified 95.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 5: 58.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{y}{x}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e+28)
   (* (/ x z) (+ 1.0 (* (* y y) -0.16666666666666666)))
   (/ (/ y (/ y x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+28) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = (y / (y / x)) / z;
	}
	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 <= 6.4d+28) then
        tmp = (x / z) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = (y / (y / x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+28) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = (y / (y / x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.4e+28:
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = (y / (y / x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+28)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(y / Float64(y / x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.4e+28)
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = (y / (y / x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+28], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000001e28

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*r/ 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in y around 0 70.2%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \cdot \frac{x}{z} \]
   5. Step-by-step derivation

      1. unpow2 70.2%

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

   6. Simplified 70.2%

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

   if 6.4000000000000001e28 < y

   1. Initial program 94.2%

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

      1. associate-*r/ 94.2%

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

      2. clear-num 89.8%

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

   3. Applied egg-rr 89.8%

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

   4. Taylor expanded in y around 0 22.6%

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

      1. clear-num 22.6%

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

      2. clear-num 22.6%

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

      3. remove-double-div 22.6%

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

      4. *-inverses 22.6%

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

      5. associate-/r* 22.3%

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

      6. associate-/r/ 22.3%

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

   6. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.3%

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

      2. *-un-lft-identity 22.3%

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

      3. associate-/l* 27.9%

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

   8. Applied egg-rr 27.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{y}{x}}}{z}\\ \end{array} \]

Alternative 6: 60.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y}{x}}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.7e-11) (/ x z) (/ (/ 1.0 (/ (/ y x) y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.7e-11) {
		tmp = x / z;
	} else {
		tmp = (1.0 / ((y / x) / y)) / z;
	}
	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 <= 3.7d-11) then
        tmp = x / z
    else
        tmp = (1.0d0 / ((y / x) / y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.7e-11) {
		tmp = x / z;
	} else {
		tmp = (1.0 / ((y / x) / y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.7e-11:
		tmp = x / z
	else:
		tmp = (1.0 / ((y / x) / y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.7e-11)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(y / x) / y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.7e-11)
		tmp = x / z;
	else
		tmp = (1.0 / ((y / x) / y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.7e-11], N[(x / z), $MachinePrecision], N[(N[(1.0 / N[(N[(y / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7000000000000001e-11

   1. Initial program 98.2%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 86.5%

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

      3. associate-/l/ 80.8%

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

      4. associate-/r/ 77.2%

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

      5. associate-/r* 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in y around 0 73.1%

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

   if 3.7000000000000001e-11 < y

   1. Initial program 94.9%

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

      1. associate-*r/ 94.9%

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

      2. clear-num 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in y around 0 23.6%

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

      1. *-un-lft-identity 23.6%

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

      2. times-frac 28.4%

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

   6. Applied egg-rr 28.4%

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

      1. associate-*l/ 28.4%

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

      2. *-un-lft-identity 28.4%

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

   8. Applied egg-rr 28.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y}{x}}{y}}}{z}\\ \end{array} \]

Alternative 7: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ x (* z (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y, double z) {
	return x / (z * (1.0 + (0.16666666666666666 * (y * y))));
}

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 * (1.0d0 + (0.16666666666666666d0 * (y * y))))
end function

Java

public static double code(double x, double y, double z) {
	return x / (z * (1.0 + (0.16666666666666666 * (y * y))));
}

Python

def code(x, y, z):
	return x / (z * (1.0 + (0.16666666666666666 * (y * y))))

Julia

function code(x, y, z)
	return Float64(x / Float64(z * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x / (z * (1.0 + (0.16666666666666666 * (y * y))));
end

Wolfram

code[x_, y_, z_] := N[(x / N[(z * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

   1. associate-/l* 96.2%

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

3. Simplified 96.2%

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

   1. clear-num 96.0%

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

   2. associate-/r/ 96.1%

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

   3. clear-num 96.1%

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

5. Applied egg-rr 96.1%

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

6. Taylor expanded in y around 0 68.7%

   \[\leadsto \frac{x}{\color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \cdot z} \]
7. Step-by-step derivation

   1. unpow2 68.7%

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

8. Simplified 68.7%

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

9. Final simplification 68.7%

   \[\leadsto \frac{x}{z \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

Alternative 8: 66.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{z}}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (/ x z) (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y, double z) {
	return (x / z) / (1.0 + (0.16666666666666666 * (y * y)));
}

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) / (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y, double z) {
	return (x / z) / (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y, z):
	return (x / z) / (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y, z)
	return Float64(Float64(x / z) / Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / z) / (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] / N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

   1. associate-/l* 96.2%

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

   2. associate-/r/ 87.5%

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

   3. associate-/l/ 83.9%

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

   4. associate-/r/ 81.1%

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

   5. associate-/r* 80.4%

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

3. Simplified 80.4%

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

   1. associate-*l/ 82.9%

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

   2. *-commutative 82.9%

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

   3. frac-times 95.8%

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

   4. clear-num 95.8%

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

   5. un-div-inv 95.8%

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

5. Applied egg-rr 95.8%

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

6. Taylor expanded in y around 0 69.4%

   \[\leadsto \frac{\frac{x}{z}}{\color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 68.7%

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

8. Simplified 69.4%

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

9. Final simplification 69.4%

   \[\leadsto \frac{\frac{x}{z}}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

Alternative 9: 59.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0001:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0001) (/ x z) (/ (* y (/ x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0001) {
		tmp = x / z;
	} else {
		tmp = (y * (x / y)) / z;
	}
	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 <= 0.0001d0) then
        tmp = x / z
    else
        tmp = (y * (x / y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0001) {
		tmp = x / z;
	} else {
		tmp = (y * (x / y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0001:
		tmp = x / z
	else:
		tmp = (y * (x / y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0001)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y * Float64(x / y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0001)
		tmp = x / z;
	else
		tmp = (y * (x / y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0001], N[(x / z), $MachinePrecision], N[(N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000005e-4

   1. Initial program 98.2%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 86.5%

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

      3. associate-/l/ 80.9%

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

      4. associate-/r/ 77.3%

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

      5. associate-/r* 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around 0 73.3%

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

   if 1.00000000000000005e-4 < y

   1. Initial program 94.8%

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

      1. associate-*r/ 94.8%

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

      2. clear-num 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in y around 0 22.2%

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

      1. *-un-lft-identity 22.2%

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

      2. times-frac 27.1%

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

   6. Applied egg-rr 27.1%

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

      1. associate-*l/ 27.1%

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

      2. *-un-lft-identity 27.1%

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

      3. associate-/r/ 27.1%

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

      4. clear-num 27.0%

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

   8. Applied egg-rr 27.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0001:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y}}{z}\\ \end{array} \]

Alternative 10: 60.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{y}{x}}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.7e-5) (/ x z) (/ (/ y (/ y x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-5) {
		tmp = x / z;
	} else {
		tmp = (y / (y / x)) / z;
	}
	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 <= 4.7d-5) then
        tmp = x / z
    else
        tmp = (y / (y / x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-5) {
		tmp = x / z;
	} else {
		tmp = (y / (y / x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.7e-5:
		tmp = x / z
	else:
		tmp = (y / (y / x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.7e-5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y / Float64(y / x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.7e-5)
		tmp = x / z;
	else
		tmp = (y / (y / x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.7e-5], N[(x / z), $MachinePrecision], N[(N[(y / N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.69999999999999972e-5

   1. Initial program 98.2%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 86.5%

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

      3. associate-/l/ 80.9%

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

      4. associate-/r/ 77.3%

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

      5. associate-/r* 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around 0 73.3%

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

   if 4.69999999999999972e-5 < y

   1. Initial program 94.8%

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

      1. associate-*r/ 94.8%

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

      2. clear-num 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in y around 0 22.5%

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

      1. clear-num 22.5%

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

      2. clear-num 22.5%

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

      3. remove-double-div 22.5%

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

      4. *-inverses 22.5%

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

      5. associate-/r* 22.2%

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

      6. associate-/r/ 22.2%

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

   6. Applied egg-rr 22.2%

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

      1. associate-*l/ 22.2%

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

      2. *-un-lft-identity 22.2%

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

      3. associate-/l* 27.1%

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

   8. Applied egg-rr 27.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{y}{x}}}{z}\\ \end{array} \]

Alternative 11: 57.9% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{z}{x}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (/ z x)))

C

double code(double x, double y, double z) {
	return 1.0 / (z / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (z / x)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (z / x);
}

Python

def code(x, y, z):
	return 1.0 / (z / x)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(z / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (z / x);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(z / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

   1. associate-/l* 96.2%

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

3. Simplified 96.2%

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

   1. clear-num 96.0%

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

   2. associate-/r/ 96.1%

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

   3. clear-num 96.1%

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

5. Applied egg-rr 96.1%

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

   1. clear-num 95.8%

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

   2. associate-/r/ 96.0%

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

   3. associate-/r* 96.4%

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

   4. clear-num 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in y around 0 62.2%

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

   1. associate-*l/ 62.4%

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

   2. associate-/l* 62.4%

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

10. Applied egg-rr 62.4%

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

11. Final simplification 62.4%

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

Alternative 12: 58.0% 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 97.5%

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

   1. associate-/l* 96.2%

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

   2. associate-/r/ 87.5%

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

   3. associate-/l/ 83.9%

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

   4. associate-/r/ 81.1%

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

   5. associate-/r* 80.4%

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

3. Simplified 80.4%

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

4. Taylor expanded in y around 0 62.4%

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

5. Final simplification 62.4%

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

Developer target: 99.7% 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 2023263 
(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))