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

Percentage Accurate: 95.8% → 99.9%

Time: 10.8s

Alternatives: 11

Speedup: 5.0×

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: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

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 / fma(x, y, (exp(z) * -1.1283791670955126)));
	}
	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 / fma(x, y, Float64(exp(z) * -1.1283791670955126))));
	end
	return 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[(x * y + N[(N[Exp[z], $MachinePrecision] * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.4%

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

      1. remove-double-neg 97.4%

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

      2. distribute-frac-neg 97.4%

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

      3. unsub-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. distribute-neg-frac2 97.4%

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

      6. neg-sub0 97.4%

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

      7. associate--r- 97.4%

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

      8. neg-sub0 97.4%

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

      9. +-commutative 97.4%

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

      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
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x + (1.0d0 / (((exp(z) * 1.1283791670955126d0) - (x * y)) / 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 + (1.0 / (((Math.exp(z) * 1.1283791670955126) - (x * y)) / 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 + (1.0 / (((math.exp(z) * 1.1283791670955126) - (x * y)) / 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(1.0 / Float64(Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y)) / 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 + (1.0 / (((exp(z) * 1.1283791670955126) - (x * y)) / 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[(1.0 / N[(N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.4%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

      3. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

      2. *-commutative 97.3%

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

   6. Simplified 97.3%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 3: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x 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 / ((exp(z) * 1.1283791670955126) - (x * 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 / ((exp(z) * 1.1283791670955126d0) - (x * 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 / ((Math.exp(z) * 1.1283791670955126) - (x * 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 / ((math.exp(z) * 1.1283791670955126) - (x * 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(Float64(exp(z) * 1.1283791670955126) - Float64(x * 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 / ((exp(z) * 1.1283791670955126) - (x * 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[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x - \frac{y}{x \cdot y - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 1.00000002)
   (- x (/ y (- (* x y) 1.1283791670955126)))
   (- x (* (/ y (exp z)) -0.8862269254527579))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 1.00000002) {
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	} else {
		tmp = x - ((y / exp(z)) * -0.8862269254527579);
	}
	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) <= 1.00000002d0) then
        tmp = x - (y / ((x * y) - 1.1283791670955126d0))
    else
        tmp = x - ((y / exp(z)) * (-0.8862269254527579d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 1.00000002) {
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	} else {
		tmp = x - ((y / Math.exp(z)) * -0.8862269254527579);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 1.00000002:
		tmp = x - (y / ((x * y) - 1.1283791670955126))
	else:
		tmp = x - ((y / math.exp(z)) * -0.8862269254527579)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 1.00000002)
		tmp = Float64(x - Float64(y / Float64(Float64(x * y) - 1.1283791670955126)));
	else
		tmp = Float64(x - Float64(Float64(y / exp(z)) * -0.8862269254527579));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 1.00000002)
		tmp = x - (y / ((x * y) - 1.1283791670955126));
	else
		tmp = x - ((y / exp(z)) * -0.8862269254527579);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000002], N[(x - N[(y / N[(N[(x * y), $MachinePrecision] - 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision] * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 1.0000000200000001

   1. Initial program 95.4%

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

      1. remove-double-neg 95.4%

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

      2. distribute-frac-neg 95.4%

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

      3. unsub-neg 95.4%

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

      4. distribute-frac-neg 95.4%

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

      5. distribute-neg-frac2 95.4%

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

      6. neg-sub0 95.3%

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

      7. associate--r- 95.3%

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

      8. neg-sub0 95.5%

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

      9. +-commutative 95.5%

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

      10. fma-define 95.5%

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

      11. *-commutative 95.5%

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

      12. distribute-rgt-neg-in 95.5%

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

      13. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in z around 0 87.6%

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

   if 1.0000000200000001 < (exp.f64 z)

   1. Initial program 92.2%

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

      1. remove-double-neg 92.2%

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

      2. distribute-frac-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. distribute-frac-neg 92.2%

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

      5. distribute-neg-frac2 92.2%

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

      6. neg-sub0 92.2%

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

      7. associate--r- 92.2%

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

      8. neg-sub0 92.2%

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

      9. +-commutative 92.2%

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

      10. fma-define 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto x - \color{blue}{-0.8862269254527579 \cdot \frac{y}{e^{z}}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x - \color{blue}{\frac{y}{e^{z}} \cdot -0.8862269254527579} \]

   7. Simplified 100.0%

      \[\leadsto x - \color{blue}{\frac{y}{e^{z}} \cdot -0.8862269254527579} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x - \frac{y}{x \cdot y - 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot -0.5641895835477563\right) - x \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -820000.0)
   (- x (/ 1.0 x))
   (+
    x
    (/
     y
     (+
      1.1283791670955126
      (- (* z (- 1.1283791670955126 (* z -0.5641895835477563))) (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * 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 (z <= (-820000.0d0)) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x + (y / (1.1283791670955126d0 + ((z * (1.1283791670955126d0 - (z * (-0.5641895835477563d0)))) - (x * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -820000.0:
		tmp = x - (1.0 / x)
	else:
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -820000.0)
		tmp = Float64(x - Float64(1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(Float64(z * Float64(1.1283791670955126 - Float64(z * -0.5641895835477563))) - Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -820000.0)
		tmp = x - (1.0 / x);
	else
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -820000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * N[(1.1283791670955126 - N[(z * -0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e5

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -8.2e5 < z

   1. Initial program 97.4%

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

      1. remove-double-neg 97.4%

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

      2. distribute-frac-neg 97.4%

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

      3. unsub-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. distribute-neg-frac2 97.4%

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

      6. neg-sub0 97.4%

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

      7. associate--r- 97.4%

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

      8. neg-sub0 97.4%

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

      9. +-commutative 97.4%

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

      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 95.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot -0.5641895835477563\right) - x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{x \cdot y - 1.1283791670955126}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -820000.0)
   (- x (/ 1.0 x))
   (+ x (/ -1.0 (/ (- (* x y) 1.1283791670955126) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (-1.0 / (((x * y) - 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 (z <= (-820000.0d0)) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x + ((-1.0d0) / (((x * y) - 1.1283791670955126d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (-1.0 / (((x * y) - 1.1283791670955126) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -820000.0:
		tmp = x - (1.0 / x)
	else:
		tmp = x + (-1.0 / (((x * y) - 1.1283791670955126) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -820000.0)
		tmp = Float64(x - Float64(1.0 / x));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(x * y) - 1.1283791670955126) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -820000.0)
		tmp = x - (1.0 / x);
	else
		tmp = x + (-1.0 / (((x * y) - 1.1283791670955126) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -820000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(N[(x * y), $MachinePrecision] - 1.1283791670955126), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e5

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -8.2e5 < z

   1. Initial program 97.4%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

      3. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

      2. *-commutative 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in z around 0 88.6%

      \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126} - y \cdot x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{x \cdot y - 1.1283791670955126}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -820000.0)
   (- x (/ 1.0 x))
   (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - 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 <= (-820000.0d0)) then
        tmp = x - (1.0d0 / x)
    else
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -820000.0) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -820000.0:
		tmp = x - (1.0 / x)
	else:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -820000.0)
		tmp = Float64(x - Float64(1.0 / x));
	else
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -820000.0)
		tmp = x - (1.0 / x);
	else
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -820000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e5

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -8.2e5 < z

   1. Initial program 97.4%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

      3. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

      2. *-commutative 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in z around 0 88.6%

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

   8. Taylor expanded in y around 0 88.6%

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

      1. associate-*r* 88.6%

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

      2. neg-mul-1 88.6%

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

      3. cancel-sign-sub-inv 88.6%

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

      4. *-commutative 88.6%

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

      5. div-sub 88.6%

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

      6. *-commutative 88.6%

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

      7. associate-*r/ 88.6%

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

      8. *-inverses 88.6%

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

      9. *-rgt-identity 88.6%

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

   10. Simplified 88.6%

       \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126}{y} - x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -820000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)} \end{array} \]

FPCore

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

C

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

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 - ((x * y) + (z * (-1.1283791670955126d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. remove-double-neg 94.6%

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

   2. distribute-frac-neg 94.6%

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

   3. unsub-neg 94.6%

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

   4. distribute-frac-neg 94.6%

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

   5. distribute-neg-frac2 94.6%

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

   6. neg-sub0 94.5%

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

   7. associate--r- 94.5%

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

   8. neg-sub0 94.7%

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

   9. +-commutative 94.7%

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

   10. fma-define 96.6%

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

   11. *-commutative 96.6%

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

   12. distribute-rgt-neg-in 96.6%

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

   13. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 83.0%

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

6. Final simplification 83.0%

   \[\leadsto x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)} \]
7. Add Preprocessing

Alternative 9: 81.9% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{x \cdot y - 1.1283791670955126} \end{array} \]

FPCore

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

C

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

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 / ((x * y) - 1.1283791670955126d0))
end function

Java

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

Python

def code(x, y, z):
	return x - (y / ((x * y) - 1.1283791670955126))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. remove-double-neg 94.6%

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

   2. distribute-frac-neg 94.6%

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

   3. unsub-neg 94.6%

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

   4. distribute-frac-neg 94.6%

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

   5. distribute-neg-frac2 94.6%

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

   6. neg-sub0 94.5%

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

   7. associate--r- 94.5%

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

   8. neg-sub0 94.7%

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

   9. +-commutative 94.7%

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

   10. fma-define 96.6%

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

   11. *-commutative 96.6%

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

   12. distribute-rgt-neg-in 96.6%

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

   13. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in z around 0 82.3%

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

6. Final simplification 82.3%

   \[\leadsto x - \frac{y}{x \cdot y - 1.1283791670955126} \]
7. Add Preprocessing

Alternative 10: 68.4% accurate, 22.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - (1.0 / 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 / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in y around inf 70.6%

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

4. Final simplification 70.6%

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

Alternative 11: 68.9% 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 94.6%

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

3. Taylor expanded in x around inf 67.2%

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

4. Final simplification 67.2%

   \[\leadsto x \]
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 2024066 
(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)))))