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

Percentage Accurate: 95.8% → 99.5%

Time: 13.1s

Alternatives: 16

Speedup: 5.8×

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: 95.8% 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: 99.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 195:\\ \;\;\;\;x + \frac{1}{\frac{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right) - x \cdot y}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (if (<= z 195.0)
     (+
      x
      (/
       1.0
       (/
        (-
         (+
          1.1283791670955126
          (* z (+ 1.1283791670955126 (* z 0.5641895835477563))))
         (* x y))
        y)))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 195.0) {
		tmp = x + (1.0 / (((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)) / y));
	} else {
		tmp = 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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 195.0d0) then
        tmp = x + (1.0d0 / (((1.1283791670955126d0 + (z * (1.1283791670955126d0 + (z * 0.5641895835477563d0)))) - (x * y)) / y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 195.0) {
		tmp = x + (1.0 / (((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)) / y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	elif z <= 195.0:
		tmp = x + (1.0 / (((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)) / y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 195.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 + Float64(z * Float64(1.1283791670955126 + Float64(z * 0.5641895835477563)))) - Float64(x * y)) / y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 195.0)
		tmp = x + (1.0 / (((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)) / y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 195.0], N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 + N[(z * N[(1.1283791670955126 + N[(z * 0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z < 195

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}} \]

      2. inv-pow 99.8%

         \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}\right)}^{-1}} \]

      3. *-commutative 99.8%

         \[\leadsto x + {\left(\frac{1.1283791670955126 \cdot e^{z} - \color{blue}{y \cdot x}}{y}\right)}^{-1} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 99.8%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}}} \]

      2. *-commutative 99.8%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{e^{z} \cdot 1.1283791670955126} - y \cdot x}{y}} \]

   6. Simplified 99.8%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}} \]

   7. Taylor expanded in z around 0 99.5%

      \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + 0.5641895835477563 \cdot z\right)\right)} - y \cdot x}{y}} \]
   8. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto x + \frac{1}{\frac{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + \color{blue}{z \cdot 0.5641895835477563}\right)\right) - y \cdot x}{y}} \]

   9. Simplified 99.5%

      \[\leadsto x + \frac{1}{\frac{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right)} - y \cdot x}{y}} \]

   if 195 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 195:\\ \;\;\;\;x + \frac{1}{\frac{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right) - x \cdot y}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{z} \cdot 1.1283791670955126 - x \cdot y\\ \mathbf{if}\;x + \frac{y}{t\_0} \leq 5 \cdot 10^{+194}:\\ \;\;\;\;x + \frac{1}{\frac{t\_0}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (exp z) 1.1283791670955126) (* x y))))
   (if (<= (+ x (/ y t_0)) 5e+194) (+ x (/ 1.0 (/ t_0 y))) (- x (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = (exp(z) * 1.1283791670955126) - (x * y);
	double tmp;
	if ((x + (y / t_0)) <= 5e+194) {
		tmp = x + (1.0 / (t_0 / y));
	} else {
		tmp = x - (1.0 / 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 = (exp(z) * 1.1283791670955126d0) - (x * y)
    if ((x + (y / t_0)) <= 5d+194) then
        tmp = x + (1.0d0 / (t_0 / y))
    else
        tmp = x - (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.exp(z) * 1.1283791670955126) - (x * y);
	double tmp;
	if ((x + (y / t_0)) <= 5e+194) {
		tmp = x + (1.0 / (t_0 / y));
	} else {
		tmp = x - (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.exp(z) * 1.1283791670955126) - (x * y)
	tmp = 0
	if (x + (y / t_0)) <= 5e+194:
		tmp = x + (1.0 / (t_0 / y))
	else:
		tmp = x - (1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))
	tmp = 0.0
	if (Float64(x + Float64(y / t_0)) <= 5e+194)
		tmp = Float64(x + Float64(1.0 / Float64(t_0 / y)));
	else
		tmp = Float64(x - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (exp(z) * 1.1283791670955126) - (x * y);
	tmp = 0.0;
	if ((x + (y / t_0)) <= 5e+194)
		tmp = x + (1.0 / (t_0 / y));
	else
		tmp = x - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], 5e+194], N[(x + N[(1.0 / N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]

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)))) < 4.99999999999999989e194

   1. Initial program 99.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}} \]

      2. inv-pow 99.7%

         \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}\right)}^{-1}} \]

      3. *-commutative 99.7%

         \[\leadsto x + {\left(\frac{1.1283791670955126 \cdot e^{z} - \color{blue}{y \cdot x}}{y}\right)}^{-1} \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 99.7%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}}} \]

      2. *-commutative 99.7%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{e^{z} \cdot 1.1283791670955126} - y \cdot x}{y}} \]

   6. Simplified 99.7%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}} \]

   if 4.99999999999999989e194 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{+194}:\\ \;\;\;\;x + \frac{1}{\frac{e^{z} \cdot 1.1283791670955126 - x \cdot y}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 5e+194) t_0 (- x (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+194) {
		tmp = t_0;
	} else {
		tmp = x - (1.0 / 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 = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 5d+194) then
        tmp = t_0
    else
        tmp = x - (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+194) {
		tmp = t_0;
	} else {
		tmp = x - (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 5e+194:
		tmp = t_0
	else:
		tmp = x - (1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 5e+194)
		tmp = t_0;
	else
		tmp = Float64(x - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 5e+194)
		tmp = t_0;
	else
		tmp = x - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+194], t$95$0, N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]

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)))) < 4.99999999999999989e194

   1. Initial program 99.6%

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

   if 4.99999999999999989e194 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{+194}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Step-by-step derivation

   1. remove-double-neg 97.7%

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

   2. distribute-frac-neg 97.7%

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

   3. unsub-neg 97.7%

      \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

   4. distribute-frac-neg 97.7%

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

   5. distribute-neg-frac2 97.7%

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

   6. neg-sub0 97.8%

      \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   7. associate--r- 97.8%

      \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

   8. neg-sub0 97.8%

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

   9. +-commutative 97.8%

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

   10. fma-define 98.6%

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

   11. *-commutative 98.6%

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

   12. distribute-rgt-neg-in 98.6%

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

   13. metadata-eval 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
4. Add Preprocessing

5. Final simplification 98.6%

   \[\leadsto x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)} \]
6. Add Preprocessing

Alternative 5: 97.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x + (y / (1.1283791670955126d0 - (y * (x + ((z * ((z * ((z * (-0.18806319451591877d0)) - 0.5641895835477563d0)) - 1.1283791670955126d0)) / y)))))
    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 = x - (1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 - (y * (x + ((z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126)) / y)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x - (1.0 / x)
	else:
		tmp = x + (y / (1.1283791670955126 - (y * (x + ((z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126)) / y)))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x - (1.0 / x);
	else
		tmp = x + (y / (1.1283791670955126 - (y * (x + ((z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126)) / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 95.6%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. unsub-neg 98.7%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 98.7%

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

      5. distribute-neg-frac2 98.7%

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

      6. neg-sub0 98.7%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 98.7%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 98.7%

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

      9. +-commutative 98.7%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.5%

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

   6. Taylor expanded in y around inf 99.1%

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

4. Final simplification 99.4%

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

Alternative 6: 86.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126 - z \cdot -1.1283791670955126}\\ \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-263}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ 1.0 x)))
        (t_1 (+ x (/ y (- 1.1283791670955126 (* z -1.1283791670955126))))))
   (if (<= z -1550000000.0)
     t_0
     (if (<= z 4.5e-263)
       t_1
       (if (<= z 2.2e-167) t_0 (if (<= z 8.2e-10) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double t_1 = x + (y / (1.1283791670955126 - (z * -1.1283791670955126)));
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 4.5e-263) {
		tmp = t_1;
	} else if (z <= 2.2e-167) {
		tmp = t_0;
	} else if (z <= 8.2e-10) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = x - (1.0d0 / x)
    t_1 = x + (y / (1.1283791670955126d0 - (z * (-1.1283791670955126d0))))
    if (z <= (-1550000000.0d0)) then
        tmp = t_0
    else if (z <= 4.5d-263) then
        tmp = t_1
    else if (z <= 2.2d-167) then
        tmp = t_0
    else if (z <= 8.2d-10) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double t_1 = x + (y / (1.1283791670955126 - (z * -1.1283791670955126)));
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 4.5e-263) {
		tmp = t_1;
	} else if (z <= 2.2e-167) {
		tmp = t_0;
	} else if (z <= 8.2e-10) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (1.0 / x)
	t_1 = x + (y / (1.1283791670955126 - (z * -1.1283791670955126)))
	tmp = 0
	if z <= -1550000000.0:
		tmp = t_0
	elif z <= 4.5e-263:
		tmp = t_1
	elif z <= 2.2e-167:
		tmp = t_0
	elif z <= 8.2e-10:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(z * -1.1283791670955126))))
	tmp = 0.0
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 4.5e-263)
		tmp = t_1;
	elseif (z <= 2.2e-167)
		tmp = t_0;
	elseif (z <= 8.2e-10)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (1.0 / x);
	t_1 = x + (y / (1.1283791670955126 - (z * -1.1283791670955126)));
	tmp = 0.0;
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 4.5e-263)
		tmp = t_1;
	elseif (z <= 2.2e-167)
		tmp = t_0;
	elseif (z <= 8.2e-10)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(1.1283791670955126 - N[(z * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1550000000.0], t$95$0, If[LessEqual[z, 4.5e-263], t$95$1, If[LessEqual[z, 2.2e-167], t$95$0, If[LessEqual[z, 8.2e-10], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e9 or 4.4999999999999997e-263 < z < 2.2e-167

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf 97.9%

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

   if -1.55e9 < z < 4.4999999999999997e-263 or 2.2e-167 < z < 8.1999999999999996e-10

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.6%

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

   6. Taylor expanded in y around 0 82.1%

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

   if 8.1999999999999996e-10 < z

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf 98.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-263}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - z \cdot -1.1283791670955126}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-167}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - z \cdot -1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-218}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-6)
   x
   (if (<= x -1.36e-226)
     (/ -1.0 x)
     (if (<= x 1.4e-218)
       (* y 0.8862269254527579)
       (if (<= x 3.35e-149) x (if (<= x 5.6e-14) (/ -1.0 x) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-6) {
		tmp = x;
	} else if (x <= -1.36e-226) {
		tmp = -1.0 / x;
	} else if (x <= 1.4e-218) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 3.35e-149) {
		tmp = x;
	} else if (x <= 5.6e-14) {
		tmp = -1.0 / x;
	} else {
		tmp = 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 <= (-1d-6)) then
        tmp = x
    else if (x <= (-1.36d-226)) then
        tmp = (-1.0d0) / x
    else if (x <= 1.4d-218) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 3.35d-149) then
        tmp = x
    else if (x <= 5.6d-14) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-6) {
		tmp = x;
	} else if (x <= -1.36e-226) {
		tmp = -1.0 / x;
	} else if (x <= 1.4e-218) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 3.35e-149) {
		tmp = x;
	} else if (x <= 5.6e-14) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-6:
		tmp = x
	elif x <= -1.36e-226:
		tmp = -1.0 / x
	elif x <= 1.4e-218:
		tmp = y * 0.8862269254527579
	elif x <= 3.35e-149:
		tmp = x
	elif x <= 5.6e-14:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-6)
		tmp = x;
	elseif (x <= -1.36e-226)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.4e-218)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 3.35e-149)
		tmp = x;
	elseif (x <= 5.6e-14)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e-6)
		tmp = x;
	elseif (x <= -1.36e-226)
		tmp = -1.0 / x;
	elseif (x <= 1.4e-218)
		tmp = y * 0.8862269254527579;
	elseif (x <= 3.35e-149)
		tmp = x;
	elseif (x <= 5.6e-14)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e-6], x, If[LessEqual[x, -1.36e-226], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.4e-218], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 3.35e-149], x, If[LessEqual[x, 5.6e-14], N[(-1.0 / x), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999955e-7 or 1.40000000000000004e-218 < x < 3.3499999999999998e-149 or 5.6000000000000001e-14 < x

   1. Initial program 97.8%

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

   3. Taylor expanded in x around inf 95.1%

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

   if -9.99999999999999955e-7 < x < -1.35999999999999992e-226 or 3.3499999999999998e-149 < x < 5.6000000000000001e-14

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf 62.8%

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

   4. Taylor expanded in x around 0 62.8%

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

   if -1.35999999999999992e-226 < x < 1.40000000000000004e-218

   1. Initial program 96.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 96.5%

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

      2. distribute-frac-neg 96.5%

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

      3. unsub-neg 96.5%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 96.5%

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

      5. distribute-neg-frac2 96.5%

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

      6. neg-sub0 97.0%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 97.0%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 97.0%

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

      9. +-commutative 97.0%

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

      10. fma-define 97.0%

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

      11. *-commutative 97.0%

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

      12. distribute-rgt-neg-in 97.0%

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

      13. metadata-eval 97.0%

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 84.7%

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

   6. Taylor expanded in y around inf 84.7%

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

   7. Taylor expanded in y around inf 65.2%

      \[\leadsto x - \frac{y}{\color{blue}{x \cdot y} - 1.1283791670955126} \]
   8. Step-by-step derivation

      1. *-commutative 65.2%

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

   9. Simplified 65.2%

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

   10. Taylor expanded in x around 0 57.2%

       \[\leadsto \color{blue}{0.8862269254527579 \cdot y} \]
   11. Step-by-step derivation

       1. *-commutative 57.2%

          \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]

   12. Simplified 57.2%

       \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-218}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 61:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot \left(z \cdot -0.5641895835477563 - 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (if (<= z 61.0)
     (+
      x
      (/
       y
       (-
        1.1283791670955126
        (+ (* x y) (* z (- (* z -0.5641895835477563) 1.1283791670955126))))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 61.0) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * ((z * -0.5641895835477563) - 1.1283791670955126)))));
	} else {
		tmp = 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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 61.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - ((x * y) + (z * ((z * (-0.5641895835477563d0)) - 1.1283791670955126d0)))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 61.0) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * ((z * -0.5641895835477563) - 1.1283791670955126)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	elif z <= 61.0:
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * ((z * -0.5641895835477563) - 1.1283791670955126)))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 61.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(Float64(x * y) + Float64(z * Float64(Float64(z * -0.5641895835477563) - 1.1283791670955126))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 61.0)
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * ((z * -0.5641895835477563) - 1.1283791670955126)))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 61.0], N[(x + N[(y / N[(1.1283791670955126 - N[(N[(x * y), $MachinePrecision] + N[(z * N[(N[(z * -0.5641895835477563), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z < 61

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.4%

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

   if 61 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 61:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot \left(z \cdot -0.5641895835477563 - 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(x \cdot y + z \cdot \left(z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right) - 1.1283791670955126\right)\right) - 1.1283791670955126}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (-
    x
    (/
     y
     (-
      (+
       (* x y)
       (*
        z
        (-
         (* z (- (* z -0.18806319451591877) 0.5641895835477563))
         1.1283791670955126)))
      1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x - (y / (((x * y) + (z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126))) - 1.1283791670955126));
	}
	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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x - (y / (((x * y) + (z * ((z * ((z * (-0.18806319451591877d0)) - 0.5641895835477563d0)) - 1.1283791670955126d0))) - 1.1283791670955126d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x - (y / (((x * y) + (z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126))) - 1.1283791670955126));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	else:
		tmp = x - (y / (((x * y) + (z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126))) - 1.1283791670955126))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(Float64(x * y) + Float64(z * Float64(Float64(z * Float64(Float64(z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126))) - 1.1283791670955126)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	else
		tmp = x - (y / (((x * y) + (z * ((z * ((z * -0.18806319451591877) - 0.5641895835477563)) - 1.1283791670955126))) - 1.1283791670955126));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(N[(z * N[(N[(z * -0.18806319451591877), $MachinePrecision] - 0.5641895835477563), $MachinePrecision]), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z

   1. Initial program 98.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. unsub-neg 98.7%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 98.7%

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

      5. distribute-neg-frac2 98.7%

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

      6. neg-sub0 98.7%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 98.7%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 98.7%

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

      9. +-commutative 98.7%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(x \cdot y + z \cdot \left(z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right) - 1.1283791670955126\right)\right) - 1.1283791670955126}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x - y \cdot -0.8862269254527579\\ \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-261}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ 1.0 x))) (t_1 (- x (* y -0.8862269254527579))))
   (if (<= z -1550000000.0)
     t_0
     (if (<= z 6.6e-261)
       t_1
       (if (<= z 4e-166) t_0 (if (<= z 2.6e-9) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double t_1 = x - (y * -0.8862269254527579);
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 6.6e-261) {
		tmp = t_1;
	} else if (z <= 4e-166) {
		tmp = t_0;
	} else if (z <= 2.6e-9) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = x - (1.0d0 / x)
    t_1 = x - (y * (-0.8862269254527579d0))
    if (z <= (-1550000000.0d0)) then
        tmp = t_0
    else if (z <= 6.6d-261) then
        tmp = t_1
    else if (z <= 4d-166) then
        tmp = t_0
    else if (z <= 2.6d-9) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double t_1 = x - (y * -0.8862269254527579);
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 6.6e-261) {
		tmp = t_1;
	} else if (z <= 4e-166) {
		tmp = t_0;
	} else if (z <= 2.6e-9) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (1.0 / x)
	t_1 = x - (y * -0.8862269254527579)
	tmp = 0
	if z <= -1550000000.0:
		tmp = t_0
	elif z <= 6.6e-261:
		tmp = t_1
	elif z <= 4e-166:
		tmp = t_0
	elif z <= 2.6e-9:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(1.0 / x))
	t_1 = Float64(x - Float64(y * -0.8862269254527579))
	tmp = 0.0
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 6.6e-261)
		tmp = t_1;
	elseif (z <= 4e-166)
		tmp = t_0;
	elseif (z <= 2.6e-9)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (1.0 / x);
	t_1 = x - (y * -0.8862269254527579);
	tmp = 0.0;
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 6.6e-261)
		tmp = t_1;
	elseif (z <= 4e-166)
		tmp = t_0;
	elseif (z <= 2.6e-9)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1550000000.0], t$95$0, If[LessEqual[z, 6.6e-261], t$95$1, If[LessEqual[z, 4e-166], t$95$0, If[LessEqual[z, 2.6e-9], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e9 or 6.5999999999999996e-261 < z < 4.00000000000000016e-166

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf 97.9%

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

   if -1.55e9 < z < 6.5999999999999996e-261 or 4.00000000000000016e-166 < z < 2.6000000000000001e-9

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.6%

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

   6. Taylor expanded in y around 0 82.1%

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

   7. Taylor expanded in z around 0 81.7%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   9. Simplified 81.7%

      \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   if 2.6000000000000001e-9 < z

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf 98.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-261}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-166}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{1}{x}\\ \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-260}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ 1.0 x))))
   (if (<= z -1550000000.0)
     t_0
     (if (<= z 2.15e-260)
       (- x (* y -0.8862269254527579))
       (if (<= z 2.9e-167)
         t_0
         (if (<= z 2.6e-9) (- x (/ y -1.1283791670955126)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 2.15e-260) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 2.9e-167) {
		tmp = t_0;
	} else if (z <= 2.6e-9) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = 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 = x - (1.0d0 / x)
    if (z <= (-1550000000.0d0)) then
        tmp = t_0
    else if (z <= 2.15d-260) then
        tmp = x - (y * (-0.8862269254527579d0))
    else if (z <= 2.9d-167) then
        tmp = t_0
    else if (z <= 2.6d-9) then
        tmp = x - (y / (-1.1283791670955126d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double tmp;
	if (z <= -1550000000.0) {
		tmp = t_0;
	} else if (z <= 2.15e-260) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 2.9e-167) {
		tmp = t_0;
	} else if (z <= 2.6e-9) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (1.0 / x)
	tmp = 0
	if z <= -1550000000.0:
		tmp = t_0
	elif z <= 2.15e-260:
		tmp = x - (y * -0.8862269254527579)
	elif z <= 2.9e-167:
		tmp = t_0
	elif z <= 2.6e-9:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(1.0 / x))
	tmp = 0.0
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 2.15e-260)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	elseif (z <= 2.9e-167)
		tmp = t_0;
	elseif (z <= 2.6e-9)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (1.0 / x);
	tmp = 0.0;
	if (z <= -1550000000.0)
		tmp = t_0;
	elseif (z <= 2.15e-260)
		tmp = x - (y * -0.8862269254527579);
	elseif (z <= 2.9e-167)
		tmp = t_0;
	elseif (z <= 2.6e-9)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1550000000.0], t$95$0, If[LessEqual[z, 2.15e-260], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e-167], t$95$0, If[LessEqual[z, 2.6e-9], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.55e9 or 2.15000000000000011e-260 < z < 2.90000000000000003e-167

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf 97.9%

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

   if -1.55e9 < z < 2.15000000000000011e-260

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.5%

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

   6. Taylor expanded in y around 0 87.6%

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

   7. Taylor expanded in z around 0 87.3%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 87.3%

         \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   9. Simplified 87.3%

      \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   if 2.90000000000000003e-167 < z < 2.6000000000000001e-9

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in y around 0 71.5%

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

   7. Taylor expanded in z around 0 71.0%

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

   if 2.6000000000000001e-9 < z

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-260}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-167}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 120:\\ \;\;\;\;x - \frac{y}{\left(x \cdot y + z \cdot -1.1283791670955126\right) - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (if (<= z 120.0)
     (- x (/ y (- (+ (* x y) (* z -1.1283791670955126)) 1.1283791670955126)))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 120.0) {
		tmp = x - (y / (((x * y) + (z * -1.1283791670955126)) - 1.1283791670955126));
	} else {
		tmp = 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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 120.0d0) then
        tmp = x - (y / (((x * y) + (z * (-1.1283791670955126d0))) - 1.1283791670955126d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 120.0) {
		tmp = x - (y / (((x * y) + (z * -1.1283791670955126)) - 1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	elif z <= 120.0:
		tmp = x - (y / (((x * y) + (z * -1.1283791670955126)) - 1.1283791670955126))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 120.0)
		tmp = Float64(x - Float64(y / Float64(Float64(Float64(x * y) + Float64(z * -1.1283791670955126)) - 1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 120.0)
		tmp = x - (y / (((x * y) + (z * -1.1283791670955126)) - 1.1283791670955126));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 120.0], N[(x - N[(y / N[(N[(N[(x * y), $MachinePrecision] + N[(z * -1.1283791670955126), $MachinePrecision]), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z < 120

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.2%

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

   if 120 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 120:\\ \;\;\;\;x - \frac{y}{\left(x \cdot y + z \cdot -1.1283791670955126\right) - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 380:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126 - x \cdot y}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (if (<= z 380.0) (+ x (/ 1.0 (/ (- 1.1283791670955126 (* x y)) y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 380.0) {
		tmp = x + (1.0 / ((1.1283791670955126 - (x * y)) / y));
	} else {
		tmp = 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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 380.0d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 - (x * y)) / y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 380.0) {
		tmp = x + (1.0 / ((1.1283791670955126 - (x * y)) / y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	elif z <= 380.0:
		tmp = x + (1.0 / ((1.1283791670955126 - (x * y)) / y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 380.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 - Float64(x * y)) / y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 380.0)
		tmp = x + (1.0 / ((1.1283791670955126 - (x * y)) / y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 380.0], N[(x + N[(1.0 / N[(N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z < 380

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}} \]

      2. inv-pow 99.8%

         \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}\right)}^{-1}} \]

      3. *-commutative 99.8%

         \[\leadsto x + {\left(\frac{1.1283791670955126 \cdot e^{z} - \color{blue}{y \cdot x}}{y}\right)}^{-1} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{{\left(\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 99.8%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - y \cdot x}{y}}} \]

      2. *-commutative 99.8%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{e^{z} \cdot 1.1283791670955126} - y \cdot x}{y}} \]

   6. Simplified 99.8%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}} \]

   7. Taylor expanded in z around 0 98.8%

      \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126} - y \cdot x}{y}} \]

   if 380 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 380:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126 - x \cdot y}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-217}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.2e-5)
   x
   (if (<= x -3.4e-183)
     (/ -1.0 x)
     (if (<= x 2.6e-217) (- x (* y -0.8862269254527579)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.2e-5) {
		tmp = x;
	} else if (x <= -3.4e-183) {
		tmp = -1.0 / x;
	} else if (x <= 2.6e-217) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = 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 <= (-2.2d-5)) then
        tmp = x
    else if (x <= (-3.4d-183)) then
        tmp = (-1.0d0) / x
    else if (x <= 2.6d-217) then
        tmp = x - (y * (-0.8862269254527579d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.2e-5) {
		tmp = x;
	} else if (x <= -3.4e-183) {
		tmp = -1.0 / x;
	} else if (x <= 2.6e-217) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.2e-5:
		tmp = x
	elif x <= -3.4e-183:
		tmp = -1.0 / x
	elif x <= 2.6e-217:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = x;
	elseif (x <= -3.4e-183)
		tmp = Float64(-1.0 / x);
	elseif (x <= 2.6e-217)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.2e-5)
		tmp = x;
	elseif (x <= -3.4e-183)
		tmp = -1.0 / x;
	elseif (x <= 2.6e-217)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.2e-5], x, If[LessEqual[x, -3.4e-183], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 2.6e-217], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e-5 or 2.59999999999999993e-217 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf 87.1%

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

   if -2.1999999999999999e-5 < x < -3.40000000000000014e-183

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 68.6%

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

   4. Taylor expanded in x around 0 68.6%

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

   if -3.40000000000000014e-183 < x < 2.59999999999999993e-217

   1. Initial program 94.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 94.8%

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

      2. distribute-frac-neg 94.8%

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

      3. unsub-neg 94.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 94.8%

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

      5. distribute-neg-frac2 94.8%

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

      6. neg-sub0 95.0%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 95.0%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 95.1%

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

      9. +-commutative 95.1%

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

      10. fma-define 95.1%

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

      11. *-commutative 95.1%

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

      12. distribute-rgt-neg-in 95.1%

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

      13. metadata-eval 95.1%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 67.6%

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

   6. Taylor expanded in y around 0 65.3%

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

   7. Taylor expanded in z around 0 59.2%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]

   9. Simplified 59.2%

      \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-217}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - \frac{y}{x \cdot y - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13200000000.0)
   (- x (/ 1.0 x))
   (if (<= z 175.0) (- x (/ y (- (* x y) 1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 175.0) {
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	} else {
		tmp = 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 <= (-13200000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 175.0d0) then
        tmp = x - (y / ((x * y) - 1.1283791670955126d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13200000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 175.0) {
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13200000000.0:
		tmp = x - (1.0 / x)
	elif z <= 175.0:
		tmp = x - (y / ((x * y) - 1.1283791670955126))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13200000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 175.0)
		tmp = Float64(x - Float64(y / Float64(Float64(x * y) - 1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13200000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 175.0)
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13200000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 175.0], N[(x - N[(y / N[(N[(x * y), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e10

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e10 < z < 175

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{x - \frac{-y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.7%

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

   if 175 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13200000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - \frac{y}{x \cdot y - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.9% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around inf 69.2%

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

4. Final simplification 69.2%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024078 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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