Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 98.0%

Time: 7.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 -5e-222) t_1 (* t_0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -5e-222) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = (x * t_0) / z
    if (t_1 <= (-5d-222)) then
        tmp = t_1
    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 t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -5e-222) {
		tmp = t_1;
	} else {
		tmp = t_0 * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = (x * t_0) / z
	tmp = 0
	if t_1 <= -5e-222:
		tmp = t_1
	else:
		tmp = t_0 * (x / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= -5e-222)
		tmp = t_1;
	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;
	t_1 = (x * t_0) / z;
	tmp = 0.0;
	if (t_1 <= -5e-222)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-222], t$95$1, N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5.00000000000000008e-222

   1. Initial program 99.8%

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

   if -5.00000000000000008e-222 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.1%

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

      1. associate-/l* 93.1%

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

      2. associate-/r/ 83.0%

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

   3. Simplified 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. frac-times 80.1%

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

      3. clear-num 80.9%

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

      4. frac-times 85.8%

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

      5. *-commutative 85.8%

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

      6. times-frac 98.7%

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

   5. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2: 75.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-6)
   (* (/ 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 <= 6e-6) {
		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 <= 6d-6) 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 <= 6e-6) {
		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 <= 6e-6:
		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 <= 6e-6)
		tmp = Float64(Float64(x / 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 <= 6e-6)
		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, 6e-6], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $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 < 6.0000000000000002e-6

   1. Initial program 96.9%

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

      1. associate-/l* 93.6%

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

      2. associate-/r/ 78.3%

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

   3. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. frac-times 75.9%

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

      3. clear-num 76.8%

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

      4. frac-times 78.7%

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

      5. *-commutative 78.7%

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

      6. times-frac 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in y around 0 68.6%

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

      1. unpow2 68.6%

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

   8. Simplified 68.6%

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

   if 6.0000000000000002e-6 < y

   1. Initial program 91.1%

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

      1. associate-*l/ 92.3%

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

      2. times-frac 87.9%

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

      3. *-commutative 87.9%

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

      4. associate-*r/ 87.9%

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

      5. *-commutative 87.9%

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

   3. Simplified 87.9%

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

4. Final simplification 73.4%

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

Alternative 3: 96.1% 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. associate-/l* 92.1%

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

   2. associate-/r/ 80.7%

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

3. Simplified 80.7%

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

   1. *-un-lft-identity 80.7%

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

   2. frac-times 79.8%

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

   3. clear-num 80.5%

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

   4. frac-times 81.0%

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

   5. *-commutative 81.0%

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

   6. times-frac 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 4: 62.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d0) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((1.0d0 / y) * ((x / z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 96.9%

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

      1. associate-*l/ 98.4%

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

      2. times-frac 78.7%

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

      3. *-commutative 78.7%

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

      4. associate-*r/ 80.0%

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

      5. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in y around 0 71.2%

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

   if 2.5 < y

   1. Initial program 91.1%

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

      1. associate-/l* 87.8%

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

      2. associate-/r/ 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in y around 0 36.3%

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

   5. Taylor expanded in y around inf 36.3%

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

      1. *-commutative 36.3%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 36.3%

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

   7. Simplified 36.3%

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

      1. associate-/r* 36.2%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{z}}{y \cdot y}} \]

      2. *-un-lft-identity 36.2%

         \[\leadsto 6 \cdot \frac{\color{blue}{1 \cdot \frac{x}{z}}}{y \cdot y} \]

      3. times-frac 36.4%

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

   9. Applied egg-rr 36.4%

      \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 5: 60.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 12.5)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (* 6.0 (* (/ 1.0 y) (/ (/ x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 12.5d0) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = 6.0d0 * ((1.0d0 / y) * ((x / z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 12.5:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 12.5)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(6.0 * Float64(Float64(1.0 / y) * Float64(Float64(x / z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 12.5)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = 6.0 * ((1.0 / y) * ((x / z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 12.5], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(1.0 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 12.5

   1. Initial program 96.9%

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

      1. associate-/l* 93.6%

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

      2. associate-/r/ 78.4%

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

   3. Simplified 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. frac-times 76.0%

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

      3. clear-num 76.9%

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

      4. frac-times 78.8%

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

      5. *-commutative 78.8%

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

      6. times-frac 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in y around 0 68.3%

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

      1. unpow2 68.3%

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

   8. Simplified 68.3%

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

   if 12.5 < y

   1. Initial program 91.0%

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

      1. associate-/l* 87.6%

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

      2. associate-/r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in y around 0 36.9%

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

   5. Taylor expanded in y around inf 36.8%

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

      1. *-commutative 36.8%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 36.8%

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

   7. Simplified 36.8%

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

      1. associate-/r* 36.7%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{z}}{y \cdot y}} \]

      2. *-un-lft-identity 36.7%

         \[\leadsto 6 \cdot \frac{\color{blue}{1 \cdot \frac{x}{z}}}{y \cdot y} \]

      3. times-frac 36.9%

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

   9. Applied egg-rr 36.9%

      \[\leadsto 6 \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\frac{1}{y} \cdot \frac{\frac{x}{z}}{y}\right)\\ \end{array} \]

Alternative 6: 60.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(6 \cdot \frac{x}{y}\right) \cdot \frac{1}{z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 12.5)
   (* (/ x z) (+ 1.0 (* -0.16666666666666666 (* y y))))
   (/ (* (* 6.0 (/ x y)) (/ 1.0 z)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = ((6.0 * (x / y)) * (1.0 / z)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 12.5d0) then
        tmp = (x / z) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = ((6.0d0 * (x / y)) * (1.0d0 / z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 12.5) {
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = ((6.0 * (x / y)) * (1.0 / z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 12.5:
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = ((6.0 * (x / y)) * (1.0 / z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 12.5)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(Float64(Float64(6.0 * Float64(x / y)) * Float64(1.0 / z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 12.5)
		tmp = (x / z) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = ((6.0 * (x / y)) * (1.0 / z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 12.5], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(6.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 12.5

   1. Initial program 96.9%

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

      1. associate-/l* 93.6%

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

      2. associate-/r/ 78.4%

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

   3. Simplified 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. frac-times 76.0%

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

      3. clear-num 76.9%

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

      4. frac-times 78.8%

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

      5. *-commutative 78.8%

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

      6. times-frac 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in y around 0 68.3%

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

      1. unpow2 68.3%

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

   8. Simplified 68.3%

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

   if 12.5 < y

   1. Initial program 91.0%

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

      1. associate-/l* 87.6%

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

      2. associate-/r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in y around 0 36.9%

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

   5. Taylor expanded in y around inf 36.8%

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

      1. *-commutative 36.8%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 36.8%

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

   7. Simplified 36.8%

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

      1. associate-/r* 36.7%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{z}}{y \cdot y}} \]

      2. div-inv 36.7%

         \[\leadsto 6 \cdot \frac{\color{blue}{x \cdot \frac{1}{z}}}{y \cdot y} \]

      3. times-frac 36.9%

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

   9. Applied egg-rr 36.9%

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

       1. associate-*r* 36.9%

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

       2. associate-*r/ 36.9%

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

   11. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{\frac{\left(6 \cdot \frac{x}{y}\right) \cdot \frac{1}{z}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.5:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(6 \cdot \frac{x}{y}\right) \cdot \frac{1}{z}}{y}\\ \end{array} \]

Alternative 7: 62.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e+71) (/ x z) (* 6.0 (/ x (* z (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+71) {
		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 <= 5.5d+71) 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 <= 5.5e+71) {
		tmp = x / z;
	} else {
		tmp = 6.0 * (x / (z * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.5e+71:
		tmp = x / z
	else:
		tmp = 6.0 * (x / (z * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e+71)
		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 <= 5.5e+71)
		tmp = x / z;
	else
		tmp = 6.0 * (x / (z * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.5e+71], 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 < 5.5e71

   1. Initial program 97.1%

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

      1. associate-*l/ 98.4%

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

      2. times-frac 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*r/ 81.2%

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

      5. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in y around 0 68.0%

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

   if 5.5e71 < y

   1. Initial program 89.1%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in y around 0 40.2%

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

   5. Taylor expanded in y around inf 40.1%

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

      1. *-commutative 40.1%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 40.1%

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

   7. Simplified 40.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 8: 62.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.6e+53) (/ x z) (* 6.0 (/ (/ x y) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e+53) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((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 <= 4.6d+53) then
        tmp = x / z
    else
        tmp = 6.0d0 * ((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 <= 4.6e+53) {
		tmp = x / z;
	} else {
		tmp = 6.0 * ((x / y) / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.6e+53:
		tmp = x / z
	else:
		tmp = 6.0 * ((x / y) / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.6e+53)
		tmp = Float64(x / z);
	else
		tmp = Float64(6.0 * Float64(Float64(x / y) / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.6e+53)
		tmp = x / z;
	else
		tmp = 6.0 * ((x / y) / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.6e+53], N[(x / z), $MachinePrecision], N[(6.0 * N[(N[(x / y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.60000000000000039e53

   1. Initial program 97.0%

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

      1. associate-*l/ 98.4%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 80.9%

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

      5. *-commutative 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in y around 0 68.9%

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

   if 4.60000000000000039e53 < y

   1. Initial program 89.7%

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

      1. associate-/l* 85.7%

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

      2. associate-/r/ 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in y around 0 38.2%

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

   5. Taylor expanded in y around inf 38.2%

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

      1. *-commutative 38.2%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 38.2%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in x around 0 38.2%

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

      1. unpow2 38.2%

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

      2. associate-*r* 38.2%

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

      3. associate-/r* 38.3%

         \[\leadsto 6 \cdot \color{blue}{\frac{\frac{x}{y}}{y \cdot z}} \]

      4. *-commutative 38.3%

         \[\leadsto 6 \cdot \frac{\frac{x}{y}}{\color{blue}{z \cdot y}} \]

   10. Simplified 38.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 9: 62.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y \cdot z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5) (/ x z) (/ (/ (* x 6.0) (* y z)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = ((x * 6.0) / (y * z)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d0) then
        tmp = x / z
    else
        tmp = ((x * 6.0d0) / (y * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5) {
		tmp = x / z;
	} else {
		tmp = ((x * 6.0) / (y * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5:
		tmp = x / z
	else:
		tmp = ((x * 6.0) / (y * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(Float64(x * 6.0) / Float64(y * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5)
		tmp = x / z;
	else
		tmp = ((x * 6.0) / (y * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5], N[(x / z), $MachinePrecision], N[(N[(N[(x * 6.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5

   1. Initial program 96.9%

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

      1. associate-*l/ 98.4%

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

      2. times-frac 78.7%

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

      3. *-commutative 78.7%

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

      4. associate-*r/ 80.0%

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

      5. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in y around 0 71.2%

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

   if 2.5 < y

   1. Initial program 91.1%

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

      1. associate-/l* 87.8%

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

      2. associate-/r/ 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in y around 0 36.3%

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

   5. Taylor expanded in y around inf 36.3%

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

      1. *-commutative 36.3%

         \[\leadsto 6 \cdot \frac{x}{\color{blue}{z \cdot {y}^{2}}} \]

      2. unpow2 36.3%

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

   7. Simplified 36.3%

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

      1. associate-*r/ 36.3%

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

      2. associate-*r* 36.3%

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

      3. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{6 \cdot x}{z \cdot y}}{y}} \]

      4. *-commutative 36.3%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot 6}}{z \cdot y}}{y} \]

      5. *-commutative 36.3%

         \[\leadsto \frac{\frac{x \cdot 6}{\color{blue}{y \cdot z}}}{y} \]

   9. Applied egg-rr 36.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 6}{y \cdot z}}{y}\\ \end{array} \]

Alternative 10: 62.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+46) (/ x z) (* y (/ x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+46) {
		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 <= 6d+46) 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 <= 6e+46) {
		tmp = x / z;
	} else {
		tmp = y * (x / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+46:
		tmp = x / z
	else:
		tmp = y * (x / (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+46)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+46)
		tmp = x / z;
	else
		tmp = y * (x / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+46], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000047e46

   1. Initial program 97.0%

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

      1. associate-*l/ 98.4%

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

      2. times-frac 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-*r/ 80.8%

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

      5. *-commutative 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in y around 0 68.8%

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

   if 6.00000000000000047e46 < y

   1. Initial program 89.8%

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

      1. associate-*r/ 89.8%

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

      2. associate-/l/ 86.1%

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

      3. *-commutative 86.1%

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

      4. times-frac 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 24.2%

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

      1. clear-num 27.6%

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

      2. un-div-inv 27.6%

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

   6. Applied egg-rr 27.6%

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

      1. div-inv 27.6%

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

      2. clear-num 24.2%

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

      3. div-inv 24.2%

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

      4. associate-*l* 38.7%

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

      5. times-frac 38.8%

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

      6. *-un-lft-identity 38.8%

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

      7. *-commutative 38.8%

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

   8. Applied egg-rr 38.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 11: 58.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. associate-*l/ 96.9%

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

   2. times-frac 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 82.0%

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

   5. *-commutative 82.0%

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

3. Simplified 82.0%

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

4. Taylor expanded in y around 0 57.5%

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

5. Final simplification 57.5%

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

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 2023207 
(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))