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

Percentage Accurate: 95.5% → 99.6%

Time: 11.0s

Alternatives: 10

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: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 4 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000002:\\ \;\;\;\;x + \frac{-1}{x + \left(-1.1283791670955126 \cdot \frac{z}{y} - \frac{1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 4e-112)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.000002)
     (+
      x
      (/
       -1.0
       (+ x (- (* -1.1283791670955126 (/ z y)) (/ 1.1283791670955126 y)))))
     (+ x (* 0.8862269254527579 (/ y (exp z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 4e-112) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.000002) {
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))));
	} else {
		tmp = x + (0.8862269254527579 * (y / exp(z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 4d-112) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.000002d0) then
        tmp = x + ((-1.0d0) / (x + (((-1.1283791670955126d0) * (z / y)) - (1.1283791670955126d0 / y))))
    else
        tmp = x + (0.8862269254527579d0 * (y / exp(z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 4e-112) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.000002) {
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))));
	} else {
		tmp = x + (0.8862269254527579 * (y / Math.exp(z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 4e-112:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.000002:
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))))
	else:
		tmp = x + (0.8862269254527579 * (y / math.exp(z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 4e-112)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.000002)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(Float64(-1.1283791670955126 * Float64(z / y)) - Float64(1.1283791670955126 / y)))));
	else
		tmp = Float64(x + Float64(0.8862269254527579 * Float64(y / exp(z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 4e-112)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.000002)
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))));
	else
		tmp = x + (0.8862269254527579 * (y / exp(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 4e-112], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.000002], N[(x + N[(-1.0 / N[(x + N[(N[(-1.1283791670955126 * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.8862269254527579 * N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 3.9999999999999998e-112

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. neg-mul-1 88.7%

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

      3. associate-/l* 88.8%

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

      4. neg-mul-1 88.8%

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

      5. associate-/r* 88.8%

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

      6. div-sub 89.1%

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

      7. metadata-eval 89.1%

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

      8. associate-/l* 89.1%

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

      9. *-commutative 89.1%

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

      10. neg-mul-1 89.1%

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

      11. distribute-lft-neg-out 89.1%

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

      12. /-rgt-identity 89.1%

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

      13. div-sub 89.0%

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

      14. associate-/r* 89.0%

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

      15. neg-mul-1 89.0%

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

      16. *-rgt-identity 89.0%

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

      17. times-frac 89.0%

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

      18. /-rgt-identity 89.0%

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

      19. *-commutative 89.0%

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

      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 3.9999999999999998e-112 < (exp.f64 z) < 1.00000200000000006

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      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%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if 1.00000200000000006 < (exp.f64 z)

   1. Initial program 93.9%

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

      1. remove-double-neg 93.9%

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

      2. neg-mul-1 93.9%

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

      3. associate-/l* 93.9%

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

      4. neg-mul-1 93.9%

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

      5. associate-/r* 93.9%

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

      6. div-sub 93.9%

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

      7. metadata-eval 93.9%

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

      8. associate-/l* 93.9%

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

      9. *-commutative 93.9%

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

      10. neg-mul-1 93.9%

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

      11. distribute-lft-neg-out 93.9%

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

      12. /-rgt-identity 93.9%

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

      13. div-sub 93.9%

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

      14. associate-/r* 93.9%

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

      15. neg-mul-1 93.9%

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

      16. *-rgt-identity 93.9%

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

      17. times-frac 93.9%

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

      18. /-rgt-identity 93.9%

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

      19. *-commutative 93.9%

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

      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 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 4 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000002:\\ \;\;\;\;x + \frac{-1}{x + \left(-1.1283791670955126 \cdot \frac{z}{y} - \frac{1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{e^{z}}\\ \end{array} \]

Alternative 2: 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 95.3%

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

   1. remove-double-neg 95.3%

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

   2. neg-mul-1 95.3%

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

   3. associate-/l* 95.3%

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

   4. neg-mul-1 95.3%

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

   5. associate-/r* 95.3%

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

   6. div-sub 95.4%

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

   7. metadata-eval 95.4%

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

   8. associate-/l* 95.4%

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

   9. *-commutative 95.4%

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

   10. neg-mul-1 95.4%

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

   11. distribute-lft-neg-out 95.4%

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

   12. /-rgt-identity 95.4%

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

   13. div-sub 95.3%

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

   14. associate-/r* 95.3%

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

   15. neg-mul-1 95.3%

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

   16. *-rgt-identity 95.3%

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

   17. times-frac 95.3%

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

   18. /-rgt-identity 95.3%

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

   19. *-commutative 95.3%

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

   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 3: 99.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{-1}{x + \left(-1.1283791670955126 \cdot \frac{z}{y} - \frac{1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -185.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.25)
     (+
      x
      (/
       -1.0
       (+ x (- (* -1.1283791670955126 (/ z y)) (/ 1.1283791670955126 y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -185.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (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 <= (-185.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.25d0) then
        tmp = x + ((-1.0d0) / (x + (((-1.1283791670955126d0) * (z / y)) - (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 <= -185.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -185.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.25:
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -185.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.25)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(Float64(-1.1283791670955126 * Float64(z / y)) - Float64(1.1283791670955126 / y)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -185.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.25)
		tmp = x + (-1.0 / (x + ((-1.1283791670955126 * (z / y)) - (1.1283791670955126 / y))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -185.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25], N[(x + N[(-1.0 / N[(x + N[(N[(-1.1283791670955126 * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -185

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. neg-mul-1 88.7%

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

      3. associate-/l* 88.8%

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

      4. neg-mul-1 88.8%

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

      5. associate-/r* 88.8%

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

      6. div-sub 89.1%

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

      7. metadata-eval 89.1%

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

      8. associate-/l* 89.1%

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

      9. *-commutative 89.1%

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

      10. neg-mul-1 89.1%

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

      11. distribute-lft-neg-out 89.1%

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

      12. /-rgt-identity 89.1%

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

      13. div-sub 89.0%

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

      14. associate-/r* 89.0%

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

      15. neg-mul-1 89.0%

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

      16. *-rgt-identity 89.0%

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

      17. times-frac 89.0%

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

      18. /-rgt-identity 89.0%

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

      19. *-commutative 89.0%

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

      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 -185 < z < 1.25

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      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 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   if 1.25 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{-1}{x + \left(-1.1283791670955126 \cdot \frac{z}{y} - \frac{1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 86.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + -0.8862269254527579 \cdot \left(y \cdot \left(-1 + z\right)\right)\\ \mathbf{if}\;z \leq -0.000205:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.059:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (* -0.8862269254527579 (* y (+ -1.0 z))))))
   (if (<= z -0.000205)
     t_0
     (if (<= z 4.2e-299)
       t_1
       (if (<= z 5.5e-214) t_0 (if (<= z 0.059) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (-0.8862269254527579 * (y * (-1.0 + z)));
	double tmp;
	if (z <= -0.000205) {
		tmp = t_0;
	} else if (z <= 4.2e-299) {
		tmp = t_1;
	} else if (z <= 5.5e-214) {
		tmp = t_0;
	} else if (z <= 0.059) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + ((-0.8862269254527579d0) * (y * ((-1.0d0) + z)))
    if (z <= (-0.000205d0)) then
        tmp = t_0
    else if (z <= 4.2d-299) then
        tmp = t_1
    else if (z <= 5.5d-214) then
        tmp = t_0
    else if (z <= 0.059d0) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (-0.8862269254527579 * (y * (-1.0 + z)));
	double tmp;
	if (z <= -0.000205) {
		tmp = t_0;
	} else if (z <= 4.2e-299) {
		tmp = t_1;
	} else if (z <= 5.5e-214) {
		tmp = t_0;
	} else if (z <= 0.059) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (-0.8862269254527579 * (y * (-1.0 + z)))
	tmp = 0
	if z <= -0.000205:
		tmp = t_0
	elif z <= 4.2e-299:
		tmp = t_1
	elif z <= 5.5e-214:
		tmp = t_0
	elif z <= 0.059:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(-0.8862269254527579 * Float64(y * Float64(-1.0 + z))))
	tmp = 0.0
	if (z <= -0.000205)
		tmp = t_0;
	elseif (z <= 4.2e-299)
		tmp = t_1;
	elseif (z <= 5.5e-214)
		tmp = t_0;
	elseif (z <= 0.059)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (-0.8862269254527579 * (y * (-1.0 + z)));
	tmp = 0.0;
	if (z <= -0.000205)
		tmp = t_0;
	elseif (z <= 4.2e-299)
		tmp = t_1;
	elseif (z <= 5.5e-214)
		tmp = t_0;
	elseif (z <= 0.059)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(-0.8862269254527579 * N[(y * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.000205], t$95$0, If[LessEqual[z, 4.2e-299], t$95$1, If[LessEqual[z, 5.5e-214], t$95$0, If[LessEqual[z, 0.059], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e-4 or 4.2000000000000002e-299 < z < 5.50000000000000024e-214

   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* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.8%

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

      7. metadata-eval 90.8%

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

      8. associate-/l* 90.8%

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

      9. *-commutative 90.8%

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

      10. neg-mul-1 90.8%

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

      11. distribute-lft-neg-out 90.8%

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

      12. /-rgt-identity 90.8%

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

      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 98.2%

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

   if -2.05e-4 < z < 4.2000000000000002e-299 or 5.50000000000000024e-214 < z < 0.058999999999999997

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.3%

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

   5. Taylor expanded in z around 0 77.1%

      \[\leadsto x + 0.8862269254527579 \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 77.1%

         \[\leadsto x + 0.8862269254527579 \cdot \left(y + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) \]

      2. neg-mul-1 77.1%

         \[\leadsto x + 0.8862269254527579 \cdot \left(y + \color{blue}{\left(-y\right)} \cdot z\right) \]

   7. Simplified 77.1%

      \[\leadsto x + 0.8862269254527579 \cdot \color{blue}{\left(y + \left(-y\right) \cdot z\right)} \]

   8. Taylor expanded in y around -inf 77.1%

      \[\leadsto x + \color{blue}{-0.8862269254527579 \cdot \left(y \cdot \left(z - 1\right)\right)} \]

   if 0.058999999999999997 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000205:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-299}:\\ \;\;\;\;x + -0.8862269254527579 \cdot \left(y \cdot \left(-1 + z\right)\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.059:\\ \;\;\;\;x + -0.8862269254527579 \cdot \left(y \cdot \left(-1 + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -175:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -175.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.25)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -175.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		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 <= (-175.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.25d0) 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 <= -175.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		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 <= -175.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.25:
		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 <= -175.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.25)
		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 <= -175.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.25)
		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, -175.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25], 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 < -175

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. neg-mul-1 88.7%

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

      3. associate-/l* 88.8%

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

      4. neg-mul-1 88.8%

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

      5. associate-/r* 88.8%

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

      6. div-sub 89.1%

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

      7. metadata-eval 89.1%

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

      8. associate-/l* 89.1%

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

      9. *-commutative 89.1%

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

      10. neg-mul-1 89.1%

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

      11. distribute-lft-neg-out 89.1%

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

      12. /-rgt-identity 89.1%

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

      13. div-sub 89.0%

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

      14. associate-/r* 89.0%

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

      15. neg-mul-1 89.0%

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

      16. *-rgt-identity 89.0%

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

      17. times-frac 89.0%

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

      18. /-rgt-identity 89.0%

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

      19. *-commutative 89.0%

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

      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 -175 < z < 1.25

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.6%

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

   if 1.25 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -175:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 86.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + y \cdot 0.8862269254527579\\ \mathbf{if}\;z \leq -0.008:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (+ x (* y 0.8862269254527579))))
   (if (<= z -0.008)
     t_0
     (if (<= z 1.55e-299)
       t_1
       (if (<= z 3.7e-211) t_0 (if (<= z 6.6e-6) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -0.008) {
		tmp = t_0;
	} else if (z <= 1.55e-299) {
		tmp = t_1;
	} else if (z <= 3.7e-211) {
		tmp = t_0;
	} else if (z <= 6.6e-6) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y * 0.8862269254527579d0)
    if (z <= (-0.008d0)) then
        tmp = t_0
    else if (z <= 1.55d-299) then
        tmp = t_1
    else if (z <= 3.7d-211) then
        tmp = t_0
    else if (z <= 6.6d-6) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -0.008) {
		tmp = t_0;
	} else if (z <= 1.55e-299) {
		tmp = t_1;
	} else if (z <= 3.7e-211) {
		tmp = t_0;
	} else if (z <= 6.6e-6) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y * 0.8862269254527579)
	tmp = 0
	if z <= -0.008:
		tmp = t_0
	elif z <= 1.55e-299:
		tmp = t_1
	elif z <= 3.7e-211:
		tmp = t_0
	elif z <= 6.6e-6:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y * 0.8862269254527579))
	tmp = 0.0
	if (z <= -0.008)
		tmp = t_0;
	elseif (z <= 1.55e-299)
		tmp = t_1;
	elseif (z <= 3.7e-211)
		tmp = t_0;
	elseif (z <= 6.6e-6)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y * 0.8862269254527579);
	tmp = 0.0;
	if (z <= -0.008)
		tmp = t_0;
	elseif (z <= 1.55e-299)
		tmp = t_1;
	elseif (z <= 3.7e-211)
		tmp = t_0;
	elseif (z <= 6.6e-6)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.008], t$95$0, If[LessEqual[z, 1.55e-299], t$95$1, If[LessEqual[z, 3.7e-211], t$95$0, If[LessEqual[z, 6.6e-6], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0080000000000000002 or 1.55e-299 < z < 3.6999999999999998e-211

   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* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.8%

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

      7. metadata-eval 90.8%

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

      8. associate-/l* 90.8%

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

      9. *-commutative 90.8%

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

      10. neg-mul-1 90.8%

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

      11. distribute-lft-neg-out 90.8%

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

      12. /-rgt-identity 90.8%

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

      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 98.2%

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

   if -0.0080000000000000002 < z < 1.55e-299 or 3.6999999999999998e-211 < z < 6.60000000000000034e-6

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.1%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 76.6%

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

      1. *-commutative 76.6%

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

   9. Simplified 76.6%

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

   if 6.60000000000000034e-6 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.008:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-211}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 86.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -0.00106:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-300}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -0.00106)
     t_0
     (if (<= z 2e-300)
       (+ x (* y 0.8862269254527579))
       (if (<= z 1.25e-212)
         t_0
         (if (<= z 1.5e-5) (+ x (/ y 1.1283791670955126)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.00106) {
		tmp = t_0;
	} else if (z <= 2e-300) {
		tmp = x + (y * 0.8862269254527579);
	} else if (z <= 1.25e-212) {
		tmp = t_0;
	} else if (z <= 1.5e-5) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-0.00106d0)) then
        tmp = t_0
    else if (z <= 2d-300) then
        tmp = x + (y * 0.8862269254527579d0)
    else if (z <= 1.25d-212) then
        tmp = t_0
    else if (z <= 1.5d-5) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.00106) {
		tmp = t_0;
	} else if (z <= 2e-300) {
		tmp = x + (y * 0.8862269254527579);
	} else if (z <= 1.25e-212) {
		tmp = t_0;
	} else if (z <= 1.5e-5) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -0.00106:
		tmp = t_0
	elif z <= 2e-300:
		tmp = x + (y * 0.8862269254527579)
	elif z <= 1.25e-212:
		tmp = t_0
	elif z <= 1.5e-5:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -0.00106)
		tmp = t_0;
	elseif (z <= 2e-300)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	elseif (z <= 1.25e-212)
		tmp = t_0;
	elseif (z <= 1.5e-5)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -0.00106)
		tmp = t_0;
	elseif (z <= 2e-300)
		tmp = x + (y * 0.8862269254527579);
	elseif (z <= 1.25e-212)
		tmp = t_0;
	elseif (z <= 1.5e-5)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00106], t$95$0, If[LessEqual[z, 2e-300], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-212], t$95$0, If[LessEqual[z, 1.5e-5], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.00105999999999999996 or 2.00000000000000005e-300 < z < 1.25000000000000011e-212

   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* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.8%

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

      7. metadata-eval 90.8%

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

      8. associate-/l* 90.8%

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

      9. *-commutative 90.8%

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

      10. neg-mul-1 90.8%

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

      11. distribute-lft-neg-out 90.8%

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

      12. /-rgt-identity 90.8%

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

      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 98.2%

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

   if -0.00105999999999999996 < z < 2.00000000000000005e-300

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   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. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 76.8%

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

      1. *-commutative 76.8%

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

   9. Simplified 76.8%

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

   if 1.25000000000000011e-212 < z < 1.50000000000000004e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.7%

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

   3. Taylor expanded in x around 0 76.4%

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

   if 1.50000000000000004e-5 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00106:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-300}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-212}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -66.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.25) (+ x (/ -1.0 (- x (/ 1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -66.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		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 <= (-66.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.25d0) 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 <= -66.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.25) {
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -66.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.25:
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -66.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.25)
		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 <= -66.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.25)
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -66.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25], 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 < -66

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. neg-mul-1 88.7%

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

      3. associate-/l* 88.8%

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

      4. neg-mul-1 88.8%

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

      5. associate-/r* 88.8%

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

      6. div-sub 89.1%

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

      7. metadata-eval 89.1%

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

      8. associate-/l* 89.1%

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

      9. *-commutative 89.1%

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

      10. neg-mul-1 89.1%

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

      11. distribute-lft-neg-out 89.1%

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

      12. /-rgt-identity 89.1%

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

      13. div-sub 89.0%

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

      14. associate-/r* 89.0%

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

      15. neg-mul-1 89.0%

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

      16. *-rgt-identity 89.0%

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

      17. times-frac 89.0%

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

      18. /-rgt-identity 89.0%

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

      19. *-commutative 89.0%

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

      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 -66 < z < 1.25

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      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 99.2%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   if 1.25 < z

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. neg-mul-1 93.8%

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

      3. associate-/l* 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.8%

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

      7. metadata-eval 93.8%

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

      8. associate-/l* 93.8%

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

      9. *-commutative 93.8%

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

      10. neg-mul-1 93.8%

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

      11. distribute-lft-neg-out 93.8%

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

      12. /-rgt-identity 93.8%

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

      13. div-sub 93.8%

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

      14. associate-/r* 93.8%

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

      15. neg-mul-1 93.8%

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

      16. *-rgt-identity 93.8%

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

      17. times-frac 93.8%

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

      18. /-rgt-identity 93.8%

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

      19. *-commutative 93.8%

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

      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 53.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -66:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.25:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 74.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.0012:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e+77) x (if (<= z 0.0012) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e+77) {
		tmp = x;
	} else if (z <= 0.0012) {
		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 <= (-9d+77)) then
        tmp = x
    else if (z <= 0.0012d0) 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 <= -9e+77) {
		tmp = x;
	} else if (z <= 0.0012) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e+77:
		tmp = x
	elif z <= 0.0012:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e+77)
		tmp = x;
	elseif (z <= 0.0012)
		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 <= -9e+77)
		tmp = x;
	elseif (z <= 0.0012)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e+77], x, If[LessEqual[z, 0.0012], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000049e77 or 0.00119999999999999989 < z

   1. Initial program 92.6%

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

      1. remove-double-neg 92.6%

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

      2. neg-mul-1 92.6%

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

      3. associate-/l* 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.8%

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

      7. metadata-eval 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. distribute-lft-neg-out 92.8%

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

      12. /-rgt-identity 92.8%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 75.2%

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

   5. Taylor expanded in x around inf 74.2%

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

   if -9.00000000000000049e77 < z < 0.00119999999999999989

   1. Initial program 97.7%

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

      1. remove-double-neg 97.7%

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

      2. neg-mul-1 97.7%

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

      3. associate-/l* 97.7%

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

      4. neg-mul-1 97.7%

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

      5. associate-/r* 97.7%

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

      6. div-sub 97.8%

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

      7. metadata-eval 97.8%

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

      8. associate-/l* 97.8%

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

      9. *-commutative 97.8%

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

      10. neg-mul-1 97.8%

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

      11. distribute-lft-neg-out 97.8%

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

      12. /-rgt-identity 97.8%

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

      13. div-sub 97.8%

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

      14. associate-/r* 97.8%

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

      15. neg-mul-1 97.8%

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

      16. *-rgt-identity 97.8%

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

      17. times-frac 97.8%

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

      18. /-rgt-identity 97.8%

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

      19. *-commutative 97.8%

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

      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 92.7%

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

      1. associate-*r/ 92.8%

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

      2. metadata-eval 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.0012:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 69.6% 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 95.3%

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

   1. remove-double-neg 95.3%

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

   2. neg-mul-1 95.3%

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

   3. associate-/l* 95.3%

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

   4. neg-mul-1 95.3%

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

   5. associate-/r* 95.3%

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

   6. div-sub 95.4%

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

   7. metadata-eval 95.4%

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

   8. associate-/l* 95.4%

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

   9. *-commutative 95.4%

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

   10. neg-mul-1 95.4%

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

   11. distribute-lft-neg-out 95.4%

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

   12. /-rgt-identity 95.4%

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

   13. div-sub 95.3%

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

   14. associate-/r* 95.3%

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

   15. neg-mul-1 95.3%

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

   16. *-rgt-identity 95.3%

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

   17. times-frac 95.3%

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

   18. /-rgt-identity 95.3%

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

   19. *-commutative 95.3%

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

   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 y around inf 68.4%

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

5. Taylor expanded in x around inf 64.9%

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

6. Final simplification 64.9%

   \[\leadsto x \]

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 2023279 
(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)))))