Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 97.5%

Time: 10.0s

Alternatives: 12

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: 96.1% 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: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \cdot t_0 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= (* x t_0) -2e-264) (/ (/ x (/ y (sin y))) 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) <= -2e-264) {
		tmp = (x / (y / sin(y))) / 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) <= (-2d-264)) then
        tmp = (x / (y / sin(y))) / 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) <= -2e-264) {
		tmp = (x / (y / Math.sin(y))) / 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) <= -2e-264:
		tmp = (x / (y / math.sin(y))) / 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(x * t_0) <= -2e-264)
		tmp = Float64(Float64(x / Float64(y / sin(y))) / 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) <= -2e-264)
		tmp = (x / (y / sin(y))) / 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[(x * t$95$0), $MachinePrecision], -2e-264], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -2e-264

   1. Initial program 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

   if -2e-264 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 92.6%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* x t_0)))
   (if (<= t_1 -2e-264) (/ t_1 z) (/ x (/ z t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -2e-264) {
		tmp = t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = x * t_0
    if (t_1 <= (-2d-264)) then
        tmp = t_1 / 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 t_1 = x * t_0;
	double tmp;
	if (t_1 <= -2e-264) {
		tmp = t_1 / z;
	} else {
		tmp = x / (z / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = x * t_0
	tmp = 0
	if t_1 <= -2e-264:
		tmp = t_1 / z
	else:
		tmp = x / (z / t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= -2e-264)
		tmp = Float64(t_1 / 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;
	t_1 = x * t_0;
	tmp = 0.0;
	if (t_1 <= -2e-264)
		tmp = t_1 / 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]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-264], N[(t$95$1 / z), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -2e-264

   1. Initial program 99.8%

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

   if -2e-264 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 92.6%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-5)
   (/ x (/ z (+ 1.0 (* -0.16666666666666666 (* y y)))))
   (* (sin y) (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-5) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} 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 <= 2.4d-5) then
        tmp = x / (z / (1.0d0 + ((-0.16666666666666666d0) * (y * y))))
    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 <= 2.4e-5) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = Math.sin(y) * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-5:
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))))
	else:
		tmp = math.sin(y) * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-5)
		tmp = Float64(x / Float64(z / Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))));
	else
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e-5)
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	else
		tmp = sin(y) * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-5], N[(x / N[(z / N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e-5

   1. Initial program 98.2%

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

      1. associate-/l* 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 70.0%

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

      1. unpow2 70.0%

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

   6. Simplified 70.0%

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

   if 2.4000000000000001e-5 < y

   1. Initial program 87.7%

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

      1. associate-/l* 96.6%

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

      2. associate-/r/ 96.6%

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

      3. associate-/l/ 87.7%

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

      4. associate-/r/ 87.6%

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

      5. associate-/r* 96.6%

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

   3. Simplified 96.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 4: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+49}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= z -1e+49) (* 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 <= -1e+49) {
		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 <= (-1d+49)) 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 <= -1e+49) {
		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 <= -1e+49:
		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 <= -1e+49)
		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 <= -1e+49)
		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[z, -1e+49], 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 < -9.99999999999999946e48

   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%

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

   if -9.99999999999999946e48 < z

   1. Initial program 94.3%

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

      1. associate-/l* 97.9%

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

   3. Simplified 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 5: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z) {
	return (Math.sin(y) / y) * (x / z);
}

Python

def code(x, y, z):
	return (math.sin(y) / y) * (x / z)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. *-commutative 95.5%

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

   2. associate-*r/ 95.7%

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

3. Simplified 95.7%

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

4. Final simplification 95.7%

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

Alternative 6: 59.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.3e+14)
   (/ x (/ z (+ 1.0 (* -0.16666666666666666 (* y y)))))
   (/ 6.0 (/ (* y (* y z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.3e+14) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = 6.0 / ((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 <= 5.3d+14) then
        tmp = x / (z / (1.0d0 + ((-0.16666666666666666d0) * (y * y))))
    else
        tmp = 6.0d0 / ((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 <= 5.3e+14) {
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = 6.0 / ((y * (y * z)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.3e+14:
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))))
	else:
		tmp = 6.0 / ((y * (y * z)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.3e+14)
		tmp = Float64(x / Float64(z / Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))));
	else
		tmp = Float64(6.0 / Float64(Float64(y * Float64(y * z)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.3e+14)
		tmp = x / (z / (1.0 + (-0.16666666666666666 * (y * y))));
	else
		tmp = 6.0 / ((y * (y * z)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.3e+14], N[(x / N[(z / N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.3e14

   1. Initial program 98.2%

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

      1. associate-/l* 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 69.9%

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

      1. unpow2 69.9%

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

   6. Simplified 69.9%

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

   if 5.3e14 < y

   1. Initial program 86.4%

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

      1. clear-num 86.3%

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

      2. un-div-inv 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y around 0 30.8%

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

      1. *-commutative 30.8%

         \[\leadsto \frac{\frac{x}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}}}{z} \]

      2. unpow2 30.8%

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

   6. Simplified 30.8%

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

   7. Taylor expanded in y around inf 30.8%

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

      1. unpow2 30.8%

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

      2. *-commutative 30.8%

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

   9. Simplified 30.8%

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

       1. clear-num 30.8%

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

       2. un-div-inv 30.8%

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

       3. associate-*r* 30.9%

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

   11. Applied egg-rr 30.9%

       \[\leadsto \color{blue}{\frac{6}{\frac{\left(z \cdot y\right) \cdot y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \]

Alternative 7: 59.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.4e+14)
   (/ (* x (+ 1.0 (* -0.16666666666666666 (* y y)))) z)
   (/ 6.0 (/ (* y (* y z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.4e+14) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = 6.0 / ((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 <= 5.4d+14) then
        tmp = (x * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))) / z
    else
        tmp = 6.0d0 / ((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 <= 5.4e+14) {
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	} else {
		tmp = 6.0 / ((y * (y * z)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.4e+14:
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z
	else:
		tmp = 6.0 / ((y * (y * z)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.4e+14)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))) / z);
	else
		tmp = Float64(6.0 / Float64(Float64(y * Float64(y * z)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.4e+14)
		tmp = (x * (1.0 + (-0.16666666666666666 * (y * y)))) / z;
	else
		tmp = 6.0 / ((y * (y * z)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.4e+14], N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(6.0 / N[(N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e14

   1. Initial program 98.2%

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

   2. Taylor expanded in y around 0 69.9%

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

      1. unpow2 69.9%

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

   4. Simplified 69.9%

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

   if 5.4e14 < y

   1. Initial program 86.4%

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

      1. clear-num 86.3%

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

      2. un-div-inv 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y around 0 30.8%

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

      1. *-commutative 30.8%

         \[\leadsto \frac{\frac{x}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}}}{z} \]

      2. unpow2 30.8%

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

   6. Simplified 30.8%

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

   7. Taylor expanded in y around inf 30.8%

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

      1. unpow2 30.8%

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

      2. *-commutative 30.8%

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

   9. Simplified 30.8%

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

       1. clear-num 30.8%

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

       2. un-div-inv 30.8%

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

       3. associate-*r* 30.9%

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

   11. Applied egg-rr 30.9%

       \[\leadsto \color{blue}{\frac{6}{\frac{\left(z \cdot y\right) \cdot y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \]

Alternative 8: 61.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.45) (/ x z) (* 6.0 (/ x (* z (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * 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 (y <= 2.45d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * (x / (z * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.45:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.45)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.45)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.45], N[(x / z), $MachinePrecision], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 98.2%

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

      1. associate-/l* 96.1%

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

      2. associate-/r/ 81.7%

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

      3. associate-/l/ 78.2%

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

      4. associate-/r/ 78.1%

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

      5. associate-/r* 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in y around 0 75.6%

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

   if 2.4500000000000002 < y

   1. Initial program 87.1%

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

      1. clear-num 87.0%

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

      2. un-div-inv 87.1%

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

   3. Applied egg-rr 87.1%

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

   4. Taylor expanded in y around 0 31.0%

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

      1. *-commutative 31.0%

         \[\leadsto \frac{\frac{x}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}}}{z} \]

      2. unpow2 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in y around inf 31.0%

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

      1. unpow2 31.0%

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

      2. *-commutative 31.0%

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

   9. Simplified 31.0%

      \[\leadsto \color{blue}{6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 9: 61.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.45) (/ x z) (/ 6.0 (/ (* y (* y z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45) {
		tmp = x / z;
	} else {
		tmp = 6.0 / ((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 <= 2.45d0) then
        tmp = x / z
    else
        tmp = 6.0d0 / ((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 <= 2.45) {
		tmp = x / z;
	} else {
		tmp = 6.0 / ((y * (y * z)) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.45:
		tmp = x / z
	else:
		tmp = 6.0 / ((y * (y * z)) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.45)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 / Float64(Float64(y * Float64(y * z)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.45)
		tmp = x / z;
	else
		tmp = 6.0 / ((y * (y * z)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.45], N[(x / z), $MachinePrecision], N[(6.0 / N[(N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000002

   1. Initial program 98.2%

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

      1. associate-/l* 96.1%

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

      2. associate-/r/ 81.7%

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

      3. associate-/l/ 78.2%

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

      4. associate-/r/ 78.1%

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

      5. associate-/r* 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in y around 0 75.6%

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

   if 2.4500000000000002 < y

   1. Initial program 87.1%

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

      1. clear-num 87.0%

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

      2. un-div-inv 87.1%

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

   3. Applied egg-rr 87.1%

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

   4. Taylor expanded in y around 0 31.0%

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

      1. *-commutative 31.0%

         \[\leadsto \frac{\frac{x}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}}}{z} \]

      2. unpow2 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in y around inf 31.0%

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

      1. unpow2 31.0%

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

      2. *-commutative 31.0%

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

   9. Simplified 31.0%

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

       1. clear-num 31.0%

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

       2. un-div-inv 31.0%

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

       3. associate-*r* 31.0%

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

   11. Applied egg-rr 31.0%

       \[\leadsto \color{blue}{\frac{6}{\frac{\left(z \cdot y\right) \cdot y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{y \cdot \left(y \cdot z\right)}{x}}\\ \end{array} \]

Alternative 10: 65.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x / (1.0 + (y * (y * 0.16666666666666666)))) / 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 / (1.0d0 + (y * (y * 0.16666666666666666d0)))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. clear-num 95.5%

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

   2. un-div-inv 95.5%

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

3. Applied egg-rr 95.5%

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

4. Taylor expanded in y around 0 70.5%

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

   1. *-commutative 70.5%

      \[\leadsto \frac{\frac{x}{1 + \color{blue}{{y}^{2} \cdot 0.16666666666666666}}}{z} \]

   2. unpow2 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in y around 0 70.5%

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

   1. unpow2 70.5%

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

   2. *-commutative 70.5%

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

   3. associate-*r* 70.5%

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

9. Simplified 70.5%

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

10. Final simplification 70.5%

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

Alternative 11: 59.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+136) (/ x z) (* (/ x y) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+136) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (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 <= 2d+136) then
        tmp = x / z
    else
        tmp = (x / y) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+136) {
		tmp = x / z;
	} else {
		tmp = (x / y) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+136:
		tmp = x / z
	else:
		tmp = (x / y) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+136)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(x / y) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+136)
		tmp = x / z;
	else
		tmp = (x / y) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+136], N[(x / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000012e136

   1. Initial program 96.7%

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

      1. associate-/l* 96.1%

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

      2. associate-/r/ 83.7%

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

      3. associate-/l/ 79.5%

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

      4. associate-/r/ 79.3%

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

      5. associate-/r* 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in y around 0 67.9%

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

   if 2.00000000000000012e136 < y

   1. Initial program 86.8%

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

      1. clear-num 86.6%

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

      2. un-div-inv 86.8%

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

   3. Applied egg-rr 86.8%

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

      1. div-inv 86.5%

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

      2. associate-/r/ 86.6%

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

      3. associate-*l* 86.7%

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

   5. Applied egg-rr 86.7%

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

   6. Taylor expanded in y around 0 28.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{z}\\ \end{array} \]

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 95.5%

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

   1. associate-/l* 96.2%

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

   2. associate-/r/ 85.2%

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

   3. associate-/l/ 80.3%

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

   4. associate-/r/ 80.2%

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

   5. associate-/r* 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in y around 0 61.5%

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

5. Final simplification 61.5%

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

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 2023279 
(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))