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

Percentage Accurate: 95.4% → 99.6%

Time: 7.9s

Alternatives: 9

Speedup: 8.5×

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: 95.4% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.00000002)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     (+ 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 if (exp(z) <= 1.00000002) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + (y / (exp(z) * 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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.00000002d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x + (y / (exp(z) * 1.1283791670955126d0))
    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 if (Math.exp(z) <= 1.00000002) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + (y / (Math.exp(z) * 1.1283791670955126));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.00000002:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x + (y / (math.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));
	elseif (exp(z) <= 1.00000002)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(exp(z) * 1.1283791670955126)));
	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);
	elseif (exp(z) <= 1.00000002)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	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], If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000002], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.0000000200000001

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in z around 0 99.9%

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

   if 1.0000000200000001 < (exp.f64 z)

   1. Initial program 93.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

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 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.8%

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

4. Final simplification 98.4%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (-1.0 / fma(exp(z), (-1.1283791670955126 / y), x));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. remove-double-neg 95.6%

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

   2. neg-mul-1 95.6%

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

   3. associate-/l* 95.7%

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

   4. neg-mul-1 95.7%

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

   5. associate-/r* 95.7%

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

   6. div-sub 95.8%

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

   7. metadata-eval 95.8%

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

   8. associate-/l* 95.8%

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

   9. *-commutative 95.8%

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

   10. neg-mul-1 95.8%

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

   11. distribute-lft-neg-out 95.8%

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

   12. /-rgt-identity 95.8%

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

   13. div-sub 95.7%

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

   14. associate-/r* 95.7%

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

   15. neg-mul-1 95.7%

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

   16. *-rgt-identity 95.7%

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

   17. times-frac 95.7%

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

   18. /-rgt-identity 95.7%

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

   19. *-commutative 95.7%

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

   20. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternative 4: 99.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -170:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -170.0)
   (+ x (/ -1.0 x))
   (if (<= z 4e-6)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -170.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * 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 <= (-170.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 4d-6) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * 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 <= -170.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -170.0:
		tmp = x + (-1.0 / x)
	elif z <= 4e-6:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -170.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 4e-6)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -170.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 4e-6)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -170.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-6], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -170

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if -170 < z < 3.99999999999999982e-6

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in z around 0 99.8%

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

   if 3.99999999999999982e-6 < z

   1. Initial program 93.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-rgt-neg-out 43.4%

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

   4. Simplified 43.4%

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

      1. expm1-log1p-u 43.1%

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

      2. expm1-udef 43.1%

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

      3. associate-/r* 43.2%

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

      4. pow1 43.2%

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

      5. pow1 43.2%

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

      6. pow-div 43.2%

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

      7. metadata-eval 43.2%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

      8. metadata-eval 43.2%

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

      9. add-sqr-sqrt 17.0%

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

      10. sqrt-unprod 43.8%

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

      11. sqr-neg 43.8%

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

      12. sqrt-prod 26.9%

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

      13. add-sqr-sqrt 43.3%

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

   6. Applied egg-rr 43.3%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 43.3%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

      2. expm1-log1p 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -170:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -260.0)
   (+ x (/ -1.0 x))
   (if (<= z 4e-6) (+ x (/ -1.0 (+ x (/ -1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -260.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / 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 <= (-260.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 4d-6) then
        tmp = x + ((-1.0d0) / (x + ((-1.1283791670955126d0) / 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 <= -260.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -260.0:
		tmp = x + (-1.0 / x)
	elif z <= 4e-6:
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -260.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 4e-6)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(-1.1283791670955126 / y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -260.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 4e-6)
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -260.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-6], N[(x + N[(-1.0 / N[(x + N[(-1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -260

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if -260 < z < 3.99999999999999982e-6

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.5%

      \[\leadsto x + \color{blue}{\frac{-1}{x - 1.1283791670955126 \cdot \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. associate-*r/ 99.5%

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

      4. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

   if 3.99999999999999982e-6 < z

   1. Initial program 93.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-rgt-neg-out 43.4%

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

   4. Simplified 43.4%

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

      1. expm1-log1p-u 43.1%

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

      2. expm1-udef 43.1%

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

      3. associate-/r* 43.2%

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

      4. pow1 43.2%

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

      5. pow1 43.2%

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

      6. pow-div 43.2%

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

      7. metadata-eval 43.2%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

      8. metadata-eval 43.2%

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

      9. add-sqr-sqrt 17.0%

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

      10. sqrt-unprod 43.8%

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

      11. sqr-neg 43.8%

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

      12. sqrt-prod 26.9%

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

      13. add-sqr-sqrt 43.3%

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

   6. Applied egg-rr 43.3%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 43.3%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

      2. expm1-log1p 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -260:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 99.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -200.0)
   (+ x (/ -1.0 x))
   (if (<= z 4e-6) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (y / (1.1283791670955126 - (x * 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 <= (-200.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 4d-6) then
        tmp = x + (y / (1.1283791670955126d0 - (x * 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 <= -200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4e-6) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -200.0:
		tmp = x + (-1.0 / x)
	elif z <= 4e-6:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -200.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 4e-6)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -200.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 4e-6)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -200.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-6], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -200

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if -200 < z < 3.99999999999999982e-6

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in z around 0 99.5%

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

   if 3.99999999999999982e-6 < z

   1. Initial program 93.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-rgt-neg-out 43.4%

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

   4. Simplified 43.4%

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

      1. expm1-log1p-u 43.1%

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

      2. expm1-udef 43.1%

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

      3. associate-/r* 43.2%

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

      4. pow1 43.2%

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

      5. pow1 43.2%

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

      6. pow-div 43.2%

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

      7. metadata-eval 43.2%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

      8. metadata-eval 43.2%

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

      9. add-sqr-sqrt 17.0%

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

      10. sqrt-unprod 43.8%

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

      11. sqr-neg 43.8%

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

      12. sqrt-prod 26.9%

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

      13. add-sqr-sqrt 43.3%

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

   6. Applied egg-rr 43.3%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 43.3%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

      2. expm1-log1p 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 73.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.15e-24) x (if (<= z 2.5e-34) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.15e-24) {
		tmp = x;
	} else if (z <= 2.5e-34) {
		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 (z <= (-3.15d-24)) then
        tmp = x
    else if (z <= 2.5d-34) 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 (z <= -3.15e-24) {
		tmp = x;
	} else if (z <= 2.5e-34) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.15e-24:
		tmp = x
	elif z <= 2.5e-34:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.15e-24)
		tmp = x;
	elseif (z <= 2.5e-34)
		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 (z <= -3.15e-24)
		tmp = x;
	elseif (z <= 2.5e-34)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.15e-24], x, If[LessEqual[z, 2.5e-34], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1499999999999999e-24 or 2.5000000000000001e-34 < z

   1. Initial program 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. distribute-rgt-neg-out 68.4%

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

   4. Simplified 68.4%

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

      1. expm1-log1p-u 59.0%

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

      2. expm1-udef 59.0%

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

      3. associate-/r* 60.8%

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

      4. pow1 60.8%

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

      5. pow1 60.8%

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

      6. pow-div 60.8%

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

      7. metadata-eval 60.8%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

      8. metadata-eval 60.8%

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

      9. add-sqr-sqrt 32.9%

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

      10. sqrt-unprod 55.9%

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

      11. sqr-neg 55.9%

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

      12. sqrt-prod 28.3%

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

      13. add-sqr-sqrt 51.4%

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

   6. Applied egg-rr 51.4%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 51.4%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

      2. expm1-log1p 52.1%

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

   8. Simplified 52.1%

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

   9. Taylor expanded in x around inf 76.6%

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

   if -3.1499999999999999e-24 < z < 2.5000000000000001e-34

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \color{blue}{\frac{-1}{x - 1.1283791670955126 \cdot \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 80.0%

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

      1. *-commutative 80.0%

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

   9. Simplified 80.0%

      \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 86.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-32}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.7e-21)
   (+ x (/ -1.0 x))
   (if (<= z 1.15e-32) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.7e-21) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-32) {
		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 (z <= (-3.7d-21)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.15d-32) 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 (z <= -3.7e-21) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-32) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.7e-21:
		tmp = x + (-1.0 / x)
	elif z <= 1.15e-32:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.7e-21)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.15e-32)
		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 (z <= -3.7e-21)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.15e-32)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.7e-21], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-32], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7000000000000002e-21

   1. Initial program 90.4%

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

      1. remove-double-neg 90.4%

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

      2. neg-mul-1 90.4%

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

      3. associate-/l* 90.6%

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

      4. neg-mul-1 90.6%

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

      5. associate-/r* 90.6%

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

      6. div-sub 90.9%

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

      7. metadata-eval 90.9%

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

      8. associate-/l* 90.9%

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

      9. *-commutative 90.9%

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

      10. neg-mul-1 90.9%

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

      11. distribute-lft-neg-out 90.9%

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

      12. /-rgt-identity 90.9%

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

      13. div-sub 90.7%

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

      14. associate-/r* 90.7%

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

      15. neg-mul-1 90.7%

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

      16. *-rgt-identity 90.7%

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

      17. times-frac 90.7%

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

      18. /-rgt-identity 90.7%

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

      19. *-commutative 90.7%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.7%

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

   if -3.7000000000000002e-21 < z < 1.15e-32

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \color{blue}{\frac{-1}{x - 1.1283791670955126 \cdot \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 80.0%

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

      1. *-commutative 80.0%

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

   9. Simplified 80.0%

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

   if 1.15e-32 < z

   1. Initial program 94.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. distribute-rgt-neg-out 46.4%

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

   4. Simplified 46.4%

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

      1. expm1-log1p-u 46.1%

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

      2. expm1-udef 46.1%

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

      3. associate-/r* 46.1%

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

      4. pow1 46.1%

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

      5. pow1 46.1%

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

      6. pow-div 46.1%

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

      7. metadata-eval 46.1%

         \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

      8. metadata-eval 46.1%

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

      9. add-sqr-sqrt 19.1%

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

      10. sqrt-unprod 46.8%

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

      11. sqr-neg 46.8%

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

      12. sqrt-prod 27.8%

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

      13. add-sqr-sqrt 45.0%

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

   6. Applied egg-rr 45.0%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 45.0%

         \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

      2. expm1-log1p 46.3%

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

   8. Simplified 46.3%

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

   9. Taylor expanded in x around inf 94.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-32}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 68.4% 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 95.6%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

2. Taylor expanded in x around inf 68.2%

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

   1. mul-1-neg 68.2%

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

   2. distribute-rgt-neg-out 68.2%

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

4. Simplified 68.2%

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

   1. expm1-log1p-u 61.2%

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

   2. expm1-udef 61.2%

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

   3. associate-/r* 62.3%

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

   4. pow1 62.3%

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

   5. pow1 62.3%

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

   6. pow-div 62.3%

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

   7. metadata-eval 62.3%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{{y}^{\color{blue}{0}}}{-x}\right)} - 1\right) \]

   8. metadata-eval 62.3%

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

   9. add-sqr-sqrt 33.9%

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

   10. sqrt-unprod 59.7%

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

   11. sqr-neg 59.7%

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

   12. sqrt-prod 28.8%

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

   13. add-sqr-sqrt 55.1%

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

6. Applied egg-rr 55.1%

   \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 55.1%

      \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

   2. expm1-log1p 55.8%

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

8. Simplified 55.8%

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

9. Taylor expanded in x around inf 74.6%

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

10. Final simplification 74.6%

    \[\leadsto x \]

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 2023268 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

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

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