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

Percentage Accurate: 95.6% → 98.2%

Time: 11.2s

Alternatives: 11

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: 95.6% 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: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

4. Final simplification 99.6%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.02:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), \mathsf{fma}\left(y, -x, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 5e-20)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.02)
     (+
      x
      (/
       y
       (fma
        z
        (fma
         z
         (fma z 0.18806319451591877 0.5641895835477563)
         1.1283791670955126)
        (fma y (- x) 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 5e-20) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.02) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), fma(y, -x, 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 5e-20)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.02)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), fma(y, Float64(-x), 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 5e-20], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.02], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 4.9999999999999999e-20

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   if 4.9999999999999999e-20 < (exp.f64 z) < 1.02

   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. associate--l+ N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), y \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{5641895835477563}{5000000000000000}\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      15. mul-1-neg N/A

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

      16. neg-lowering-neg.f64 99.8

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

   5. Simplified 99.8%

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

   if 1.02 < (exp.f64 z)

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.02:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.02)
     (+
      x
      (/
       y
       (-
        (fma
         z
         (fma z 0.5641895835477563 1.1283791670955126)
         1.1283791670955126)
        (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.02) {
		tmp = x + (y / (fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.02)
		tmp = Float64(x + Float64(y / Float64(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.02], N[(x + N[(y / N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1.02

   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. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 1.02 < (exp.f64 z)

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.02:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.02:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 5e-20)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.02)
     (+ x (/ y (fma y (- x) (fma z 1.1283791670955126 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 5e-20) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.02) {
		tmp = x + (y / fma(y, -x, fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 5e-20)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.02)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 5e-20], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.02], N[(x + N[(y / N[(y * (-x) + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 4.9999999999999999e-20

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   if 4.9999999999999999e-20 < (exp.f64 z) < 1.02

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.02 < (exp.f64 z)

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.02:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.02) (+ x (/ y (fma y (- x) 1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.02) {
		tmp = x + (y / fma(y, -x, 1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.02)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), 1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.02], N[(x + N[(y / N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1.02

   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

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 98.4

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

   5. Simplified 98.4%

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

   if 1.02 < (exp.f64 z)

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 6: 85.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-91}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.5e-91)
   (+ x (/ -1.0 x))
   (if (<= z 7.4e-23) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e-91) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.4e-23) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-9.5d-91)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 7.4d-23) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e-91) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.4e-23) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.5e-91:
		tmp = x + (-1.0 / x)
	elif z <= 7.4e-23:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.5e-91)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 7.4e-23)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.5e-91)
		tmp = x + (-1.0 / x);
	elseif (z <= 7.4e-23)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.5e-91], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.4e-23], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -9.5e-91

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 91.9

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

   5. Simplified 91.9%

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

   if -9.5e-91 < z < 7.4000000000000005e-23

   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

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 79.4%

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

   if 7.4000000000000005e-23 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.5%

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

Alternative 7: 85.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-89}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.45e-89)
   (+ x (/ -1.0 x))
   (if (<= z 8e-23) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e-89) {
		tmp = x + (-1.0 / x);
	} else if (z <= 8e-23) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.45e-89)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 8e-23)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.45e-89], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e-23], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999996e-89

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 91.9

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

   5. Simplified 91.9%

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

   if -1.44999999999999996e-89 < z < 7.99999999999999968e-23

   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

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} + x \]

      3. accelerator-lowering-fma.f64 79.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   8. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   if 7.99999999999999968e-23 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.5%

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

Alternative 8: 74.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+45)
   (/ -1.0 x)
   (if (<= z 8e-23) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+45) {
		tmp = -1.0 / x;
	} else if (z <= 8e-23) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+45)
		tmp = Float64(-1.0 / x);
	elseif (z <= 8e-23)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+45], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, 8e-23], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.19999999999999995e45

   1. Initial program 89.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. distribute-neg-frac2 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

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

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

      17. mul-1-neg N/A

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

      18. *-lowering-*.f64 N/A

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

      19. mul-1-neg N/A

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

      20. neg-lowering-neg.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 54.9

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

   8. Simplified 54.9%

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

   if -1.19999999999999995e45 < z < 7.99999999999999968e-23

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} + x \]

      3. accelerator-lowering-fma.f64 76.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   8. Simplified 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   if 7.99999999999999968e-23 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.5%

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

Alternative 9: 72.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e-67) x (if (<= x 2.75e-27) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e-67) {
		tmp = x;
	} else if (x <= 2.75e-27) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e-67)
		tmp = x;
	elseif (x <= 2.75e-27)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e-67], x, If[LessEqual[x, 2.75e-27], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000031e-67 or 2.7500000000000001e-27 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.3%

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

   if -9.00000000000000031e-67 < x < 2.7500000000000001e-27

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 66.3

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

   5. Simplified 66.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} + x \]

      3. accelerator-lowering-fma.f64 54.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   8. Simplified 54.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-152}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-183}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e-152) x (if (<= x 1.85e-183) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-152) {
		tmp = x;
	} else if (x <= 1.85e-183) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.6d-152)) then
        tmp = x
    else if (x <= 1.85d-183) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-152) {
		tmp = x;
	} else if (x <= 1.85e-183) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e-152:
		tmp = x
	elif x <= 1.85e-183:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e-152)
		tmp = x;
	elseif (x <= 1.85e-183)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e-152)
		tmp = x;
	elseif (x <= 1.85e-183)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e-152], x, If[LessEqual[x, 1.85e-183], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000006e-152 or 1.8499999999999999e-183 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 83.1%

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

   if -1.60000000000000006e-152 < x < 1.8499999999999999e-183

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 58.7

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

   5. Simplified 58.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

      \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.3% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 70.2%

   \[\leadsto \color{blue}{x} \]
6. 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 2024204 
(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)))))