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

Percentage Accurate: 95.5% → 99.8%

Time: 8.4s

Alternatives: 8

Speedup: 0.5×

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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+
      (/
       y
       (-
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right), z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{30000000000000000} \cdot z + \frac{5641895835477563}{10000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right), z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      8. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.004:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -5e+32) t_0 (if (<= t_1 0.004) (- (- x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5e+32) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = -(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_1 <= (-5d+32)) then
        tmp = t_0
    else if (t_1 <= 0.004d0) then
        tmp = -(-x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -5e+32) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = -(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_1 <= -5e+32:
		tmp = t_0
	elif t_1 <= 0.004:
		tmp = -(-x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -5e+32)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = Float64(-Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -5e+32)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = -(-x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+32], t$95$0, If[LessEqual[t$95$1, 0.004], (-(-x)), t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -4.9999999999999997e32 or 0.0040000000000000001 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -4.9999999999999997e32 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 12.0

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

   5. Applied rewrites 12.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto -\left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.004:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_0 2e+274) t_0 (+ (/ -1.0 x) x))))

C

double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+274) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if (t_0 <= 2d+274) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / x) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / ((1.1283791670955126 * Math.exp(z)) - (y * x))) + x;
	double tmp;
	if (t_0 <= 2e+274) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / ((1.1283791670955126 * math.exp(z)) - (y * x))) + x
	tmp = 0
	if t_0 <= 2e+274:
		tmp = t_0
	else:
		tmp = (-1.0 / x) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_0 <= 2e+274)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / x) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	tmp = 0.0;
	if (t_0 <= 2e+274)
		tmp = t_0;
	else
		tmp = (-1.0 / x) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+274], t$95$0, N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.99999999999999984e274

   1. Initial program 99.1%

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

   if 1.99999999999999984e274 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 53.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 2 \cdot 10^{+274}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2 < (exp.f64 z)

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y} + \color{blue}{-1 \cdot x}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot y + \left(-1 \cdot x\right) \cdot y}} \]

      4. remove-double-neg N/A

         \[\leadsto x + \frac{y}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} + \left(-1 \cdot x\right) \cdot y} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) + \left(-1 \cdot x\right) \cdot y} \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot \left(-1 \cdot y\right)\right)\right)} + \left(-1 \cdot x\right) \cdot y} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right)\right) \cdot \left(-1 \cdot y\right)} + \left(-1 \cdot x\right) \cdot y} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right) \cdot \frac{e^{z}}{y}\right)} \cdot \left(-1 \cdot y\right) + \left(-1 \cdot x\right) \cdot y} \]

      9. metadata-eval N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\frac{-5641895835477563}{5000000000000000}} \cdot \frac{e^{z}}{y}\right) \cdot \left(-1 \cdot y\right) + \left(-1 \cdot x\right) \cdot y} \]

      10. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} + \left(-1 \cdot x\right) \cdot y} \]

      11. distribute-rgt-neg-out N/A

          \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot y\right)\right)} + \left(-1 \cdot x\right) \cdot y} \]

      12. associate-*r* N/A

          \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y}\right) \cdot y\right)\right) + \color{blue}{-1 \cdot \left(x \cdot y\right)}} \]

      13. mul-1-neg N/A

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

      14. distribute-neg-in N/A

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

      15. distribute-rgt-in N/A

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

      16. *-lft-identity N/A

          \[\leadsto x + \frac{y}{\mathsf{neg}\left(y \cdot \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y} + \color{blue}{1 \cdot x}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto x + \frac{y}{\mathsf{neg}\left(y \cdot \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y} + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot x\right)\right)} \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto x + \frac{y}{\mathsf{neg}\left(y \cdot \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{y} - -1 \cdot x\right)}\right)} \]

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 6: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2 < (exp.f64 z)

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0) (+ (/ y (- 1.1283791670955126 (* y x))) x) (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = ((-1.0d0) / x) + x
    else if (exp(z) <= 2.0d0) then
        tmp = (y / (1.1283791670955126d0 - (y * x))) + x
    else
        tmp = -(-x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (Math.exp(z) <= 2.0) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = (-1.0 / x) + x
	elif math.exp(z) <= 2.0:
		tmp = (y / (1.1283791670955126 - (y * x))) + x
	else:
		tmp = -(-x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(1.1283791670955126 - Float64(y * x))) + x);
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = (-1.0 / x) + x;
	elseif (exp(z) <= 2.0)
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	else
		tmp = -(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.0%

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

   if 2 < (exp.f64 z)

   1. Initial program 92.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ -\left(-x\right) \end{array} \]

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 Float64(-Float64(-x))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-(-x))

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

   3. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

   4. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

   5. neg-sub0 N/A

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

   8. sub-neg N/A

      \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

   11. mul-1-neg N/A

       \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   12. unsub-neg N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

   13. lower--.f64 N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

   14. associate-*l/ N/A

       \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

   15. *-lft-identity N/A

       \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

   16. lower-/.f64 N/A

       \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

   17. unpow2 N/A

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

   18. lower-*.f64 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in x around inf

   \[\leadsto --1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 73.0%

   \[\leadsto -\left(-x\right) \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024244 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

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

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