Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Percentage Accurate: 95.5% → 99.8%

Time: 9.4s

Alternatives: 12

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (* (- (* (/ (exp z) x) 1.1283791670955126) y) x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / ((((exp(z) / x) * 1.1283791670955126) - y) * x)) + 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 (exp(z) <= 0.0d0) then
        tmp = ((-1.0d0) / x) + x
    else
        tmp = (y / ((((exp(z) / x) * 1.1283791670955126d0) - y) * x)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / ((((Math.exp(z) / x) * 1.1283791670955126) - y) * x)) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = (-1.0 / x) + x
	else:
		tmp = (y / ((((math.exp(z) / x) * 1.1283791670955126) - y) * x)) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(Float64(Float64(exp(z) / x) * 1.1283791670955126) - y) * x)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = (-1.0 / x) + x;
	else
		tmp = (y / ((((exp(z) / x) * 1.1283791670955126) - y) * x)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(N[Exp[z], $MachinePrecision] / x), $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \frac{y}{\color{blue}{x \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)} \cdot x} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x}} \cdot \frac{5641895835477563}{5000000000000000} - y\right) \cdot x} \]

      7. lower-exp.f64 99.8

         \[\leadsto x + \frac{y}{\left(\frac{\color{blue}{e^{z}}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4000000000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -500.0)
     t_0
     (if (<= t_1 4000000000000.0)
       (+ (/ y (fma z 1.1283791670955126 1.1283791670955126)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 4000000000000.0) {
		tmp = (y / fma(z, 1.1283791670955126, 1.1283791670955126)) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 4000000000000.0)
		tmp = Float64(Float64(y / fma(z, 1.1283791670955126, 1.1283791670955126)) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$0, If[LessEqual[t$95$1, 4000000000000.0], N[(N[(y / N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -500 or 4e12 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 92.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 92.7

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 92.7%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if -500 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4e12

   1. Initial program 99.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      7. lower-*.f64 82.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} + \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{1.1283791670955126}, 1.1283791670955126\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -500:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 4000000000000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4000000000000:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -500.0)
     t_0
     (if (<= t_1 4000000000000.0) (+ (/ y 1.1283791670955126) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 4000000000000.0) {
		tmp = (y / 1.1283791670955126) + x;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_1 <= (-500.0d0)) then
        tmp = t_0
    else if (t_1 <= 4000000000000.0d0) then
        tmp = (y / 1.1283791670955126d0) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 4000000000000.0) {
		tmp = (y / 1.1283791670955126) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_0
	elif t_1 <= 4000000000000.0:
		tmp = (y / 1.1283791670955126) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 4000000000000.0)
		tmp = Float64(Float64(y / 1.1283791670955126) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 4000000000000.0)
		tmp = (y / 1.1283791670955126) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$0, If[LessEqual[t$95$1, 4000000000000.0], N[(N[(y / 1.1283791670955126), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -500 or 4e12 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 92.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 92.7

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 92.7%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if -500 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4e12

   1. Initial program 99.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      7. lower-*.f64 82.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} + \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{1.1283791670955126}, 1.1283791670955126\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.4%

       \[\leadsto x + \frac{y}{1.1283791670955126} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -500:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 4000000000000:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, \mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), \frac{1.1283791670955126}{x}\right), z, \frac{1.1283791670955126}{x}\right) - y\right) \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (*
      (-
       (fma
        (fma
         (/ z x)
         (fma 0.18806319451591877 z 0.5641895835477563)
         (/ 1.1283791670955126 x))
        z
        (/ 1.1283791670955126 x))
       y)
      x))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / ((fma(fma((z / x), fma(0.18806319451591877, z, 0.5641895835477563), (1.1283791670955126 / x)), z, (1.1283791670955126 / x)) - y) * x)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(fma(fma(Float64(z / x), fma(0.18806319451591877, z, 0.5641895835477563), Float64(1.1283791670955126 / x)), z, Float64(1.1283791670955126 / x)) - y) * x)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(N[(z / x), $MachinePrecision] * N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] + N[(1.1283791670955126 / x), $MachinePrecision]), $MachinePrecision] * z + N[(1.1283791670955126 / x), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \frac{y}{\color{blue}{x \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)} \cdot x} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x}} \cdot \frac{5641895835477563}{5000000000000000} - y\right) \cdot x} \]

      7. lower-exp.f64 99.8

         \[\leadsto x + \frac{y}{\left(\frac{\color{blue}{e^{z}}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\left(\left(z \cdot \left(z \cdot \left(\frac{5641895835477563}{30000000000000000} \cdot \frac{z}{x} + \frac{5641895835477563}{10000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) - y\right) \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.5%

      \[\leadsto x + \frac{y}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, \mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), \frac{1.1283791670955126}{x}\right), z, \frac{1.1283791670955126}{x}\right) - y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, \mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), \frac{1.1283791670955126}{x}\right), z, \frac{1.1283791670955126}{x}\right) - y\right) \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (*
      (-
       (/
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        x)
       y)
      x))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (((fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) / x) - y) * x)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) / x) - y) * x)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \frac{y}{\color{blue}{x \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)} \cdot x} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x}} \cdot \frac{5641895835477563}{5000000000000000} - y\right) \cdot x} \]

      7. lower-exp.f64 99.8

         \[\leadsto x + \frac{y}{\left(\frac{\color{blue}{e^{z}}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\left(\left(z \cdot \left(z \cdot \left(\frac{5641895835477563}{30000000000000000} \cdot \frac{z}{x} + \frac{5641895835477563}{10000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) - y\right) \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.5%

      \[\leadsto x + \frac{y}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, \mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), \frac{1.1283791670955126}{x}\right), z, \frac{1.1283791670955126}{x}\right) - y\right) \cdot x} \]

   9. Taylor expanded in x around 0

      \[\leadsto x + \frac{y}{\left(\frac{\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)}{x} - y\right) \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto x + \frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (*
      (-
       (/
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        x)
       y)
      x))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (((fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) / x) - y) * x)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) / x) - y) * x)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \frac{y}{\color{blue}{x \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right) \cdot x}} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)} \cdot x} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x} \cdot \frac{5641895835477563}{5000000000000000}} - y\right) \cdot x} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{e^{z}}{x}} \cdot \frac{5641895835477563}{5000000000000000} - y\right) \cdot x} \]

      7. lower-exp.f64 99.8

         \[\leadsto x + \frac{y}{\left(\frac{\color{blue}{e^{z}}}{x} \cdot 1.1283791670955126 - y\right) \cdot x} \]

   5. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{e^{z}}{x} \cdot 1.1283791670955126 - y\right) \cdot x}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\left(\left(z \cdot \left(z \cdot \left(\frac{5641895835477563}{30000000000000000} \cdot \frac{z}{x} + \frac{5641895835477563}{10000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right) - y\right) \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.5%

      \[\leadsto x + \frac{y}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, \mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), \frac{1.1283791670955126}{x}\right), z, \frac{1.1283791670955126}{x}\right) - y\right) \cdot x} \]

   9. Taylor expanded in x around 0

      \[\leadsto x + \frac{y}{\left(\frac{\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)}{x} - y\right) \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto x + \frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} \]

   12. Taylor expanded in z around 0

       \[\leadsto x + \frac{y}{\left(\frac{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)}{x} - y\right) \cdot x} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.4%

       \[\leadsto x + \frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/ y (* (- (/ (fma z 1.1283791670955126 1.1283791670955126) x) y) x))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (((fma(z, 1.1283791670955126, 1.1283791670955126) / x) - y) * x)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(Float64(fma(z, 1.1283791670955126, 1.1283791670955126) / x) - y) * x)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      7. lower-*.f64 92.9

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Applied rewrites 92.9%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto x + \frac{y}{-1 \cdot \color{blue}{\left(x \cdot \left(-1 \cdot \frac{\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z}{x} - -1 \cdot y\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.7%

      \[\leadsto x + \frac{y}{\left(\frac{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}{x} - y\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}{x} - y\right) \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (-
      (fma (fma 0.5641895835477563 z 1.1283791670955126) z 1.1283791670955126)
      (* y x)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. lower-fma.f64 94.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right)}, z, 1.1283791670955126\right) - x \cdot y} \]

   5. Applied rewrites 94.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.5641895835477563 \cdot z, z, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/ y (- (fma (* 0.5641895835477563 z) z 1.1283791670955126) (* y x)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (fma((0.5641895835477563 * z), z, 1.1283791670955126) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(fma(Float64(0.5641895835477563 * z), z, 1.1283791670955126) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(0.5641895835477563 * z), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. lower-fma.f64 94.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right)}, z, 1.1283791670955126\right) - x \cdot y} \]

   5. Applied rewrites 94.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.1%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(0.5641895835477563 \cdot z, z, 1.1283791670955126\right) - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.5641895835477563 \cdot z, z, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (fma z 1.1283791670955126 (- 1.1283791670955126 (* y x)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / fma(z, 1.1283791670955126, (1.1283791670955126 - (y * x)))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / fma(z, 1.1283791670955126, Float64(1.1283791670955126 - Float64(y * x)))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(z * 1.1283791670955126 + N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      7. lower-*.f64 92.9

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Applied rewrites 92.9%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -31500:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -31500.0)
   (+ (/ -1.0 x) x)
   (if (<= z 2.6e+35)
     (+ (/ y (- 1.1283791670955126 (* y x))) x)
     (+ (/ y (* 1.1283791670955126 z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -31500.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 2.6e+35) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = (y / (1.1283791670955126 * 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 (z <= (-31500.0d0)) then
        tmp = ((-1.0d0) / x) + x
    else if (z <= 2.6d+35) then
        tmp = (y / (1.1283791670955126d0 - (y * x))) + x
    else
        tmp = (y / (1.1283791670955126d0 * z)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -31500.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 2.6e+35) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = (y / (1.1283791670955126 * z)) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -31500.0:
		tmp = (-1.0 / x) + x
	elif z <= 2.6e+35:
		tmp = (y / (1.1283791670955126 - (y * x))) + x
	else:
		tmp = (y / (1.1283791670955126 * z)) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -31500.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 2.6e+35)
		tmp = Float64(Float64(y / Float64(1.1283791670955126 - Float64(y * x))) + x);
	else
		tmp = Float64(Float64(y / Float64(1.1283791670955126 * z)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -31500.0)
		tmp = (-1.0 / x) + x;
	elseif (z <= 2.6e+35)
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	else
		tmp = (y / (1.1283791670955126 * z)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -31500.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.6e+35], N[(N[(y / N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -31500

   1. Initial program 87.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if -31500 < z < 2.60000000000000007e35

   1. Initial program 99.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000}} - x \cdot y} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]

   if 2.60000000000000007e35 < z

   1. Initial program 91.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      7. lower-*.f64 78.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Applied rewrites 78.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126 - y \cdot x\right)}} \]

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto x + \frac{y}{z \cdot \color{blue}{1.1283791670955126}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -31500:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot z} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.2% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. lower-/.f64 68.1

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

5. Applied rewrites 68.1%

   \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

6. Final simplification 68.1%

   \[\leadsto \frac{-1}{x} + x \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(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 = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))