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

Percentage Accurate: 95.5% → 99.4%

Time: 8.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.5% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+ (pow (/ (fma (- y) x 1.1283791670955126) y) -1.0) 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 = pow((fma(-y, x, 1.1283791670955126) / y), -1.0) + 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(fma(Float64(-y), x, 1.1283791670955126) / y) ^ -1.0) + 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[Power[N[(N[((-y) * x + 1.1283791670955126), $MachinePrecision] / y), $MachinePrecision], -1.0], $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.9%

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

      1. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \color{blue}{1.1283791670955126}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-y, x, \frac{5641895835477563}{5000000000000000}\right)}{y}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-y, x, \frac{5641895835477563}{5000000000000000}\right)}{y}}} \]

      4. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 90.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 40.3

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

   5. Applied rewrites 40.3%

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

   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%

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

Alternative 2: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq -5 \lor \neg \left(t\_0 \leq 0.0001\right):\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (or (<= t_0 -5.0) (not (<= t_0 0.0001))) (+ (/ -1.0 x) 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 <= -5.0) || !(t_0 <= 0.0001)) {
		tmp = (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (y / ((1.1283791670955126d0 * exp(z)) - (y * x))) + x
    if ((t_0 <= (-5.0d0)) .or. (.not. (t_0 <= 0.0001d0))) then
        tmp = ((-1.0d0) / x) + x
    else
        tmp = -(-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 <= -5.0) || !(t_0 <= 0.0001)) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = -(-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 <= -5.0) or not (t_0 <= 0.0001):
		tmp = (-1.0 / x) + x
	else:
		tmp = -(-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 <= -5.0) || !(t_0 <= 0.0001))
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(-Float64(-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 <= -5.0) || ~((t_0 <= 0.0001)))
		tmp = (-1.0 / x) + x;
	else
		tmp = -(-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[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 0.0001]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

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)))) < -5 or 1.00000000000000005e-4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

   if -5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.00000000000000005e-4

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 1.6

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -5 \lor \neg \left(\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.0001\right):\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_0 \leq 10^{+238}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_0 1e+238) 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 <= 1e+238) {
		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 <= 1d+238) 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 <= 1e+238) {
		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 <= 1e+238:
		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 <= 1e+238)
		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 <= 1e+238)
		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, 1e+238], 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)))) < 1e238

   1. Initial program 99.5%

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

   if 1e238 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 49.7%

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

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 10^{+238}:\\ \;\;\;\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+ (/ y (fma (- y) x 1.1283791670955126)) 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(-y, x, 1.1283791670955126)) + 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 / fma(Float64(-y), x, 1.1283791670955126)) + 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[((-y) * x + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.9%

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

      1. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-y, x, \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 90.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 40.3

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

   5. Applied rewrites 40.3%

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

   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%

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

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 4.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 650000:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 4.6e-163)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 650000.0)
     (+ (/ y (- 1.1283791670955126 (* y x))) x)
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 4.6e-163) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 650000.0) {
		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) <= 4.6d-163) then
        tmp = ((-1.0d0) / x) + x
    else if (exp(z) <= 650000.0d0) 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) <= 4.6e-163) {
		tmp = (-1.0 / x) + x;
	} else if (Math.exp(z) <= 650000.0) {
		tmp = (y / (1.1283791670955126 - (y * x))) + x;
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 4.6e-163:
		tmp = (-1.0 / x) + x
	elif math.exp(z) <= 650000.0:
		tmp = (y / (1.1283791670955126 - (y * x))) + x
	else:
		tmp = -(-x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 4.6e-163)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 650000.0)
		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) <= 4.6e-163)
		tmp = (-1.0 / x) + x;
	elseif (exp(z) <= 650000.0)
		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], 4.6e-163], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 650000.0], 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) < 4.5999999999999999e-163

   1. Initial program 92.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 4.5999999999999999e-163 < (exp.f64 z) < 6.5e5

   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 99.9%

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

   if 6.5e5 < (exp.f64 z)

   1. Initial program 90.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 40.3

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

   5. Applied rewrites 40.3%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 4.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 650000:\\ \;\;\;\;\frac{y}{1.1283791670955126 - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. lift--.f64 N/A

      \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 98.4

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 7: 68.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-250} \lor \neg \left(x \leq 2.4 \cdot 10^{-85}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 3.8e-250) (not (<= x 2.4e-85))) (- (- x)) (/ -1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 3.8e-250) || !(x <= 2.4e-85)) {
		tmp = -(-x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= 3.8d-250) .or. (.not. (x <= 2.4d-85))) then
        tmp = -(-x)
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 3.8e-250) || !(x <= 2.4e-85)) {
		tmp = -(-x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 3.8e-250) or not (x <= 2.4e-85):
		tmp = -(-x)
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 3.8e-250) || !(x <= 2.4e-85))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= 3.8e-250) || ~((x <= 2.4e-85)))
		tmp = -(-x);
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 3.8e-250], N[Not[LessEqual[x, 2.4e-85]], $MachinePrecision]], (-(-x)), N[(-1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.79999999999999971e-250 or 2.4000000000000001e-85 < x

   1. Initial program 96.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.9%

      \[\leadsto -\left(-x\right) \]

   if 3.79999999999999971e-250 < x < 2.4000000000000001e-85

   1. Initial program 94.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 29.2

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-250} \lor \neg \left(x \leq 2.4 \cdot 10^{-85}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% 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 96.0%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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 59.7

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 69.6%

   \[\leadsto -\left(-x\right) \]
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 2024271 
(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)))))