Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 98.5%

Time: 7.5s

Alternatives: 12

Speedup: 1.0×

• 64.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+226}\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+226)))
     (* y (/ (/ (cosh x) x) z))
     (/ (/ y x) z))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e+226)) {
		tmp = y * ((cosh(x) / x) / z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e+226)) {
		tmp = y * ((Math.cosh(x) / x) / z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e+226):
		tmp = y * ((math.cosh(x) / x) / z)
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e+226))
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / z));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e+226)))
		tmp = y * ((cosh(x) / x) / z);
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e+226]], $MachinePrecision]], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -inf.0 or 9.99999999999999961e225 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.2%

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

      1. associate-*l/ 78.2%

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

   3. Simplified 78.2%

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

      1. associate-/r/ 65.0%

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

      2. associate-/l* 67.2%

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

      3. *-commutative 67.2%

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

      4. expm1-log1p-u 36.1%

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

      5. expm1-udef 36.1%

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

      6. associate-/l* 41.4%

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

      7. times-frac 48.8%

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

   5. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 48.8%

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

      2. expm1-log1p 88.9%

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

      3. associate-*r/ 94.1%

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

      4. *-commutative 94.1%

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

      5. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

   if -inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.99999999999999961e225

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -\infty \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{+226}\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 2: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate-*l/ 83.8%

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

3. Simplified 83.8%

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

   1. associate-/r/ 74.0%

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

   2. associate-/l* 71.1%

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

   3. *-commutative 71.1%

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

   4. expm1-log1p-u 43.4%

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

   5. expm1-udef 35.5%

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

   6. associate-/l* 39.4%

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

   7. times-frac 46.9%

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

5. Applied egg-rr 46.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 54.0%

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

   2. expm1-log1p 90.8%

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

   3. associate-*r/ 95.5%

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

   4. *-commutative 95.5%

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

   5. associate-*r/ 94.6%

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

7. Simplified 94.6%

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

   1. associate-/r* 77.5%

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

   2. *-commutative 77.5%

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

   3. associate-*l/ 78.1%

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

   4. *-commutative 78.1%

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

   5. associate-/r* 99.1%

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

   6. *-commutative 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 68.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (+ 1.0 (* x (* x 0.5))) (* x z)))))
   (if (<= x -2.2e-12)
     t_0
     (if (<= x 1.2e-102)
       (/ (/ y z) x)
       (if (<= x 2.1e+121)
         (/ (+ (/ y x) (* 0.5 (* y x))) z)
         (if (<= x 9.5e+237) t_0 (* (* y x) (/ 0.5 z))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	double tmp;
	if (x <= -2.2e-12) {
		tmp = t_0;
	} else if (x <= 1.2e-102) {
		tmp = (y / z) / x;
	} else if (x <= 2.1e+121) {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	} else if (x <= 9.5e+237) {
		tmp = t_0;
	} else {
		tmp = (y * x) * (0.5 / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((1.0d0 + (x * (x * 0.5d0))) / (x * z))
    if (x <= (-2.2d-12)) then
        tmp = t_0
    else if (x <= 1.2d-102) then
        tmp = (y / z) / x
    else if (x <= 2.1d+121) then
        tmp = ((y / x) + (0.5d0 * (y * x))) / z
    else if (x <= 9.5d+237) then
        tmp = t_0
    else
        tmp = (y * x) * (0.5d0 / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	double tmp;
	if (x <= -2.2e-12) {
		tmp = t_0;
	} else if (x <= 1.2e-102) {
		tmp = (y / z) / x;
	} else if (x <= 2.1e+121) {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	} else if (x <= 9.5e+237) {
		tmp = t_0;
	} else {
		tmp = (y * x) * (0.5 / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((1.0 + (x * (x * 0.5))) / (x * z))
	tmp = 0
	if x <= -2.2e-12:
		tmp = t_0
	elif x <= 1.2e-102:
		tmp = (y / z) / x
	elif x <= 2.1e+121:
		tmp = ((y / x) + (0.5 * (y * x))) / z
	elif x <= 9.5e+237:
		tmp = t_0
	else:
		tmp = (y * x) * (0.5 / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(x * z)))
	tmp = 0.0
	if (x <= -2.2e-12)
		tmp = t_0;
	elseif (x <= 1.2e-102)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 2.1e+121)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(y * x))) / z);
	elseif (x <= 9.5e+237)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * x) * Float64(0.5 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	tmp = 0.0;
	if (x <= -2.2e-12)
		tmp = t_0;
	elseif (x <= 1.2e-102)
		tmp = (y / z) / x;
	elseif (x <= 2.1e+121)
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	elseif (x <= 9.5e+237)
		tmp = t_0;
	else
		tmp = (y * x) * (0.5 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e-12], t$95$0, If[LessEqual[x, 1.2e-102], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e+121], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 9.5e+237], t$95$0, N[(N[(y * x), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.19999999999999992e-12 or 2.1000000000000002e121 < x < 9.50000000000000061e237

   1. Initial program 73.2%

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

      1. associate-*l/ 73.2%

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

   3. Simplified 73.2%

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

      1. associate-/r/ 56.7%

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

      2. associate-/l* 58.8%

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

      3. *-commutative 58.8%

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

      4. expm1-log1p-u 31.8%

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

      5. expm1-udef 31.8%

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

      6. associate-/l* 38.0%

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

      7. times-frac 47.3%

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

   5. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 47.3%

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

      2. expm1-log1p 85.6%

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

      3. associate-*r/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 44.4%

      \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 44.4%

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)} \]

      2. associate-/l/ 44.4%

         \[\leadsto y \cdot \left(\color{blue}{\frac{\frac{1}{z}}{x}} + 0.5 \cdot \frac{x}{z}\right) \]

      3. associate-*r/ 44.4%

         \[\leadsto y \cdot \left(\frac{\frac{1}{z}}{x} + \color{blue}{\frac{0.5 \cdot x}{z}}\right) \]

      4. frac-add 53.4%

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{z} \cdot z + x \cdot \left(0.5 \cdot x\right)}{x \cdot z}} \]

      5. *-commutative 53.4%

         \[\leadsto y \cdot \frac{\frac{1}{z} \cdot z + x \cdot \color{blue}{\left(x \cdot 0.5\right)}}{x \cdot z} \]

   10. Applied egg-rr 53.4%

       \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{z} \cdot z + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}} \]

   11. Taylor expanded in z around 0 53.4%

       \[\leadsto y \cdot \frac{\color{blue}{1} + x \cdot \left(x \cdot 0.5\right)}{x \cdot z} \]

   if -2.19999999999999992e-12 < x < 1.2e-102

   1. Initial program 87.0%

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

      1. associate-*l/ 86.8%

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

   3. Simplified 86.8%

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

      1. associate-/r/ 86.9%

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

      2. associate-/l* 87.4%

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

      3. *-commutative 87.4%

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

      4. expm1-log1p-u 57.0%

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

      5. expm1-udef 42.7%

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

      6. associate-/l* 42.7%

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

      7. times-frac 47.7%

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

   5. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 60.6%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 86.8%

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

      4. *-commutative 86.8%

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

      5. associate-*r/ 85.4%

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

   7. Simplified 85.4%

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

      1. associate-/r* 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*l/ 87.5%

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

      4. *-commutative 87.5%

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

      5. associate-/r* 98.5%

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

      6. *-commutative 98.5%

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

   9. Applied egg-rr 98.5%

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

   10. Taylor expanded in x around 0 87.5%

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

       1. associate-/l/ 98.5%

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

   12. Simplified 98.5%

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

   if 1.2e-102 < x < 2.1000000000000002e121

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 56.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   if 9.50000000000000061e237 < x

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 74.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 74.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 74.3%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate-/l* 74.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{z}{y \cdot x}}} \]

      2. associate-/r/ 74.3%

         \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(y \cdot x\right)} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(y \cdot x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+237}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \end{array} \]

Alternative 4: 65.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \mathbf{if}\;x \leq -65:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+100} \lor \neg \left(x \leq 5.4 \cdot 10^{+237}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y x) (/ 0.5 z))))
   (if (<= x -65.0)
     t_0
     (if (<= x 1.4)
       (/ (/ y z) x)
       (if (or (<= x 1.95e+100) (not (<= x 5.4e+237)))
         t_0
         (* 0.5 (/ y (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * x) * (0.5 / z);
	double tmp;
	if (x <= -65.0) {
		tmp = t_0;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if ((x <= 1.95e+100) || !(x <= 5.4e+237)) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * (0.5d0 / z)
    if (x <= (-65.0d0)) then
        tmp = t_0
    else if (x <= 1.4d0) then
        tmp = (y / z) / x
    else if ((x <= 1.95d+100) .or. (.not. (x <= 5.4d+237))) then
        tmp = t_0
    else
        tmp = 0.5d0 * (y / (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * x) * (0.5 / z);
	double tmp;
	if (x <= -65.0) {
		tmp = t_0;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if ((x <= 1.95e+100) || !(x <= 5.4e+237)) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y / (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * x) * (0.5 / z)
	tmp = 0
	if x <= -65.0:
		tmp = t_0
	elif x <= 1.4:
		tmp = (y / z) / x
	elif (x <= 1.95e+100) or not (x <= 5.4e+237):
		tmp = t_0
	else:
		tmp = 0.5 * (y / (z / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * x) * Float64(0.5 / z))
	tmp = 0.0
	if (x <= -65.0)
		tmp = t_0;
	elseif (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	elseif ((x <= 1.95e+100) || !(x <= 5.4e+237))
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * x) * (0.5 / z);
	tmp = 0.0;
	if (x <= -65.0)
		tmp = t_0;
	elseif (x <= 1.4)
		tmp = (y / z) / x;
	elseif ((x <= 1.95e+100) || ~((x <= 5.4e+237)))
		tmp = t_0;
	else
		tmp = 0.5 * (y / (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -65.0], t$95$0, If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, 1.95e+100], N[Not[LessEqual[x, 5.4e+237]], $MachinePrecision]], t$95$0, N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -65 or 1.3999999999999999 < x < 1.95e100 or 5.3999999999999999e237 < x

   1. Initial program 81.5%

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

   2. Taylor expanded in x around 0 40.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 40.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 40.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 40.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

   5. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate-/l* 40.0%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{z}{y \cdot x}}} \]

      2. associate-/r/ 40.0%

         \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(y \cdot x\right)} \]

   7. Applied egg-rr 40.0%

      \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(y \cdot x\right)} \]

   if -65 < x < 1.3999999999999999

   1. Initial program 89.8%

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

      1. associate-*l/ 89.7%

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

   3. Simplified 89.7%

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

      1. associate-/r/ 89.8%

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

      2. associate-/l* 89.3%

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

      3. *-commutative 89.3%

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

      4. expm1-log1p-u 62.5%

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

      5. expm1-udef 44.5%

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

      6. associate-/l* 44.5%

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

      7. times-frac 49.2%

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

   5. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 65.3%

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

      2. expm1-log1p 97.7%

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

      3. associate-*r/ 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*r/ 87.7%

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

   7. Simplified 87.7%

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

      1. associate-/r* 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-*l/ 89.3%

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

      4. *-commutative 89.3%

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

      5. associate-/r* 97.9%

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

      6. *-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in x around 0 89.3%

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

       1. associate-/l/ 97.8%

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

   12. Simplified 97.8%

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

   if 1.95e100 < x < 5.3999999999999999e237

   1. Initial program 68.0%

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

   2. Taylor expanded in x around 0 35.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 35.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 35.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 35.2%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

   5. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{z}} \]

   6. Taylor expanded in y around 0 35.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ 35.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 35.2%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      3. associate-*r/ 35.2%

         \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{z}} \]

      4. associate-/l* 54.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{\frac{z}{x}}} \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{\frac{z}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+100} \lor \neg \left(x \leq 5.4 \cdot 10^{+237}\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 5: 65.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-123)
   (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z))))
   (/ (+ (/ y x) (* 0.5 (* y x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-123) {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	} else {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-123)) then
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    else
        tmp = ((y / x) + (0.5d0 * (y * x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-123) {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	} else {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-123:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	else:
		tmp = ((y / x) + (0.5 * (y * x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-123)
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(y * x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-123)
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	else
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-123], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000001e-123

   1. Initial program 88.0%

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

      1. associate-*l/ 88.0%

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

   3. Simplified 88.0%

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

      1. associate-/r/ 81.6%

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

      2. associate-/l* 83.9%

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

      3. *-commutative 83.9%

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

      4. expm1-log1p-u 51.2%

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

      5. expm1-udef 44.9%

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

      6. associate-/l* 44.9%

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

      7. times-frac 53.4%

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

   5. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 59.7%

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

      2. expm1-log1p 97.7%

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

      3. associate-*r/ 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 72.4%

      \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]

   if -2.0000000000000001e-123 < y

   1. Initial program 81.4%

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

   2. Taylor expanded in x around 0 59.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 6: 61.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -65 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -65.0) (not (<= x 6200.0)))
   (* 0.5 (* x (/ y z)))
   (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -65.0) || !(x <= 6200.0)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (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 ((x <= (-65.0d0)) .or. (.not. (x <= 6200.0d0))) then
        tmp = 0.5d0 * (x * (y / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -65.0) || !(x <= 6200.0)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -65.0) or not (x <= 6200.0):
		tmp = 0.5 * (x * (y / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -65.0) || !(x <= 6200.0))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -65.0) || ~((x <= 6200.0)))
		tmp = 0.5 * (x * (y / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -65.0], N[Not[LessEqual[x, 6200.0]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -65 or 6200 < x

   1. Initial program 79.0%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 39.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 39.4%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

   5. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate-/l* 39.4%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{z}{y \cdot x}}} \]

      2. div-inv 39.4%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\frac{z}{y \cdot x}}} \]

   7. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\frac{z}{y \cdot x}}} \]
   8. Step-by-step derivation

      1. associate-/r* 29.5%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{\frac{z}{y}}{x}}} \]

      2. associate-/r/ 29.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{\frac{z}{y}} \cdot x\right)} \]

      3. clear-num 29.5%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{y}{z}} \cdot x\right) \]

   9. Applied egg-rr 29.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{y}{z} \cdot x\right)} \]

   if -65 < x < 6200

   1. Initial program 89.9%

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

      1. associate-*l/ 89.8%

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

   3. Simplified 89.8%

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

      1. associate-/r/ 89.0%

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

      2. associate-/l* 88.5%

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

      3. *-commutative 88.5%

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

      4. expm1-log1p-u 62.0%

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

      5. expm1-udef 44.1%

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

      6. associate-/l* 44.1%

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

      7. times-frac 48.7%

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

   5. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 64.7%

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

      2. expm1-log1p 96.8%

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

      3. associate-*r/ 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*r/ 87.8%

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

   7. Simplified 87.8%

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

      1. associate-/r* 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*l/ 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-/r* 97.9%

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

      6. *-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in x around 0 88.5%

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

       1. associate-/l/ 97.0%

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

   12. Simplified 97.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 7: 64.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -65 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -65.0) (not (<= x 6200.0)))
   (* 0.5 (/ y (/ z x)))
   (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -65.0) || !(x <= 6200.0)) {
		tmp = 0.5 * (y / (z / x));
	} else {
		tmp = (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 ((x <= (-65.0d0)) .or. (.not. (x <= 6200.0d0))) then
        tmp = 0.5d0 * (y / (z / x))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -65.0) || !(x <= 6200.0)) {
		tmp = 0.5 * (y / (z / x));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -65.0) or not (x <= 6200.0):
		tmp = 0.5 * (y / (z / x))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -65.0) || !(x <= 6200.0))
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -65.0) || ~((x <= 6200.0)))
		tmp = 0.5 * (y / (z / x));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -65.0], N[Not[LessEqual[x, 6200.0]], $MachinePrecision]], N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -65 or 6200 < x

   1. Initial program 79.0%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 39.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 39.4%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

   5. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{z}} \]

   6. Taylor expanded in y around 0 39.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}} \]

      2. *-commutative 39.4%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z} \]

      3. associate-*r/ 39.4%

         \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{z}} \]

      4. associate-/l* 36.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{\frac{z}{x}}} \]

   8. Simplified 36.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{\frac{z}{x}}} \]

   if -65 < x < 6200

   1. Initial program 89.9%

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

      1. associate-*l/ 89.8%

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

   3. Simplified 89.8%

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

      1. associate-/r/ 89.0%

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

      2. associate-/l* 88.5%

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

      3. *-commutative 88.5%

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

      4. expm1-log1p-u 62.0%

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

      5. expm1-udef 44.1%

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

      6. associate-/l* 44.1%

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

      7. times-frac 48.7%

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

   5. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 64.7%

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

      2. expm1-log1p 96.8%

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

      3. associate-*r/ 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*r/ 87.8%

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

   7. Simplified 87.8%

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

      1. associate-/r* 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*l/ 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-/r* 97.9%

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

      6. *-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in x around 0 88.5%

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

       1. associate-/l/ 97.0%

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

   12. Simplified 97.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 8: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* y (+ (* x 0.5) (/ 1.0 x))) z))

C

double code(double x, double y, double z) {
	return (y * ((x * 0.5) + (1.0 / 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 = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (y * ((x * 0.5) + (1.0 / x))) / z;
}

Python

def code(x, y, z):
	return (y * ((x * 0.5) + (1.0 / x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in x around 0 61.3%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

3. Taylor expanded in y around 0 61.3%

   \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]

4. Final simplification 61.3%

   \[\leadsto \frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z} \]

Alternative 9: 65.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ (/ y x) (* 0.5 (* y x))) z))

C

double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (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 = ((y / x) + (0.5d0 * (y * x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (y * x))) / z;
}

Python

def code(x, y, z):
	return ((y / x) + (0.5 * (y * x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(y * x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y / x) + (0.5 * (y * x))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in x around 0 61.3%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

3. Final simplification 61.3%

   \[\leadsto \frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z} \]

Alternative 10: 50.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-66) (/ y (* x z)) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-66) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-66)) then
        tmp = y / (x * z)
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-66) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-66:
		tmp = y / (x * z)
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-66)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-66)
		tmp = y / (x * z);
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-66], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e-66

   1. Initial program 87.8%

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

      1. associate-*l/ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around 0 51.3%

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

   if -2e-66 < y

   1. Initial program 81.9%

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

   2. Taylor expanded in x around 0 46.6%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 11: 49.2% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate-*l/ 83.8%

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

3. Simplified 83.8%

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

4. Taylor expanded in x around 0 43.9%

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

5. Final simplification 43.9%

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

Alternative 12: 52.7% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate-*l/ 83.8%

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

3. Simplified 83.8%

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

   1. associate-/r/ 74.0%

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

   2. associate-/l* 71.1%

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

   3. *-commutative 71.1%

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

   4. expm1-log1p-u 43.4%

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

   5. expm1-udef 35.5%

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

   6. associate-/l* 39.4%

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

   7. times-frac 46.9%

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

5. Applied egg-rr 46.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cosh x}{x} \cdot \frac{y}{z}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 54.0%

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

   2. expm1-log1p 90.8%

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

   3. associate-*r/ 95.5%

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

   4. *-commutative 95.5%

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

   5. associate-*r/ 94.6%

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

7. Simplified 94.6%

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

   1. associate-/r* 77.5%

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

   2. *-commutative 77.5%

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

   3. associate-*l/ 78.1%

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

   4. *-commutative 78.1%

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

   5. associate-/r* 99.1%

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

   6. *-commutative 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in x around 0 43.9%

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

    1. associate-/l/ 51.4%

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

12. Simplified 51.4%

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

13. Final simplification 51.4%

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

Developer target: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = 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 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023305 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))