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

Percentage Accurate: 95.6% → 99.5%

Time: 9.1s

Alternatives: 14

Speedup: 2.8×

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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.000005:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.000005)
     (-
      x
      (/
       y
       (fma
        x
        y
        (fma
         (fma
          (fma -0.18806319451591877 z -0.5641895835477563)
          z
          -1.1283791670955126)
         z
         -1.1283791670955126))))
     (fma (/ 0.8862269254527579 (exp z)) y 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) <= 1.000005) {
		tmp = x - (y / fma(x, y, fma(fma(fma(-0.18806319451591877, z, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126)));
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 1.000005)
		tmp = Float64(x - Float64(y / fma(x, y, fma(fma(fma(-0.18806319451591877, z, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126))));
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.000005], N[(x - N[(y / N[(x * y + N[(N[(N[(-0.18806319451591877 * z + -0.5641895835477563), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1.00000500000000003

   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

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)\right) - \frac{5641895835477563}{5000000000000000}}}{y}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.8

          \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right)}, z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}{y}} \]

   7. Applied rewrites 99.8%

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}}{y}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num-rev N/A

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

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

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

      7. clear-num-rev N/A

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

      8. lift-/.f64 N/A

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

   9. Applied rewrites 99.9%

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

   if 1.00000500000000003 < (exp.f64 z)

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{5000000000000000}{5641895835477563} \cdot 1}}{e^{z}} \cdot y + x \]

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.000005:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}}\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (- x (/ y (- (* y x) (* 1.1283791670955126 (exp z)))))))
   (if (<= t_1 -100.0)
     t_0
     (if (<= t_1 5e-19) (+ (/ y 1.1283791670955126) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = x - (y / ((y * x) - (1.1283791670955126 * exp(z))));
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-19) {
		tmp = (y / 1.1283791670955126) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = x - (y / ((y * x) - (1.1283791670955126d0 * exp(z))))
    if (t_1 <= (-100.0d0)) then
        tmp = t_0
    else if (t_1 <= 5d-19) then
        tmp = (y / 1.1283791670955126d0) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = x - (y / ((y * x) - (1.1283791670955126 * Math.exp(z))));
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-19) {
		tmp = (y / 1.1283791670955126) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = x - (y / ((y * x) - (1.1283791670955126 * math.exp(z))))
	tmp = 0
	if t_1 <= -100.0:
		tmp = t_0
	elif t_1 <= 5e-19:
		tmp = (y / 1.1283791670955126) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(x - Float64(y / Float64(Float64(y * x) - Float64(1.1283791670955126 * exp(z)))))
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_0;
	elseif (t_1 <= 5e-19)
		tmp = Float64(Float64(y / 1.1283791670955126) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = x - (y / ((y * x) - (1.1283791670955126 * exp(z))));
	tmp = 0.0;
	if (t_1 <= -100.0)
		tmp = t_0;
	elseif (t_1 <= 5e-19)
		tmp = (y / 1.1283791670955126) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y / N[(N[(y * x), $MachinePrecision] - N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$0, If[LessEqual[t$95$1, 5e-19], N[(N[(y / 1.1283791670955126), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -100 or 5.0000000000000004e-19 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -100 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.0000000000000004e-19

   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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.1%

      \[\leadsto x + \frac{y}{1.1283791670955126} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}} \leq -100:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}}\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.8862269254527579 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (- x (/ y (- (* y x) (* 1.1283791670955126 (exp z)))))))
   (if (<= t_1 -100.0)
     t_0
     (if (<= t_1 5e-19) (+ (* 0.8862269254527579 y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = x - (y / ((y * x) - (1.1283791670955126 * exp(z))));
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-19) {
		tmp = (0.8862269254527579 * y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = x - (y / ((y * x) - (1.1283791670955126d0 * exp(z))))
    if (t_1 <= (-100.0d0)) then
        tmp = t_0
    else if (t_1 <= 5d-19) then
        tmp = (0.8862269254527579d0 * y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = x - (y / ((y * x) - (1.1283791670955126 * Math.exp(z))));
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-19) {
		tmp = (0.8862269254527579 * y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = x - (y / ((y * x) - (1.1283791670955126 * math.exp(z))))
	tmp = 0
	if t_1 <= -100.0:
		tmp = t_0
	elif t_1 <= 5e-19:
		tmp = (0.8862269254527579 * y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(x - Float64(y / Float64(Float64(y * x) - Float64(1.1283791670955126 * exp(z)))))
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_0;
	elseif (t_1 <= 5e-19)
		tmp = Float64(Float64(0.8862269254527579 * y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = x - (y / ((y * x) - (1.1283791670955126 * exp(z))));
	tmp = 0.0;
	if (t_1 <= -100.0)
		tmp = t_0;
	elseif (t_1 <= 5e-19)
		tmp = (0.8862269254527579 * y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y / N[(N[(y * x), $MachinePrecision] - N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$0, If[LessEqual[t$95$1, 5e-19], N[(N[(0.8862269254527579 * y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -100 or 5.0000000000000004e-19 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -100 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.0000000000000004e-19

   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

      \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto x + \left(\color{blue}{\frac{\frac{-5641895835477563}{5000000000000000} \cdot \left(y \cdot z\right)}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto x + \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}} \leq -100:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot e^{z}} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;0.8862269254527579 \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 1.2)
     (- x (/ y (fma x y -1.1283791670955126)))
     (- x (/ y (- (* y x) (* 1.1283791670955126 z)))))))

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) <= 1.2) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = x - (y / ((y * x) - (1.1283791670955126 * z)));
	}
	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) <= 1.2)
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(y * x) - Float64(1.1283791670955126 * z))));
	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], 1.2], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(y * x), $MachinePrecision] - N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1.19999999999999996

   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

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{x \cdot y - \frac{5641895835477563}{5000000000000000}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 1.19999999999999996 < (exp.f64 z)

   1. Initial program 95.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

4. Final simplification 98.8%

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

Alternative 6: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)\right) - \frac{5641895835477563}{5000000000000000}}}{y}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 93.4

          \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right)}, z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}{y}} \]

   7. Applied rewrites 93.4%

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}}{y}} \]

   8. Taylor expanded in x around inf

      \[\leadsto x + \frac{-1}{\frac{x \cdot \color{blue}{\left(\left(y + \frac{z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)}{x}\right) - \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right)}}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto x + \frac{-1}{\frac{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}{x} + y\right) \cdot \color{blue}{x}}{y}} \]

   12. Applied rewrites 96.7%

       \[\leadsto \color{blue}{x - \frac{y}{\left(y + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, -0.18806319451591877, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}{x}\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

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

Alternative 7: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)\right) - \frac{5641895835477563}{5000000000000000}}}{y}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 93.4

          \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right)}, z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}{y}} \]

   7. Applied rewrites 93.4%

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}}{y}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num-rev N/A

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

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

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

      7. clear-num-rev N/A

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

      8. lift-/.f64 N/A

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

   9. Applied rewrites 93.4%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

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

Alternative 8: 96.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

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

Alternative 9: 96.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.3%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(0.5641895835477563 \cdot z, z, 1.1283791670955126\right) - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

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

Alternative 10: 97.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{y - y \cdot z}{\mathsf{fma}\left(-x, y, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, -0.18806319451591877, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -17000000.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.25e+14)
     (+ (/ (- y (* y z)) (fma (- x) y 1.1283791670955126)) x)
     (-
      x
      (/
       y
       (fma
        (fma
         (fma z -0.18806319451591877 -0.5641895835477563)
         z
         -1.1283791670955126)
        z
        -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -17000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.25e+14) {
		tmp = ((y - (y * z)) / fma(-x, y, 1.1283791670955126)) + x;
	} else {
		tmp = x - (y / fma(fma(fma(z, -0.18806319451591877, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.25e+14)
		tmp = Float64(Float64(Float64(y - Float64(y * z)) / fma(Float64(-x), y, 1.1283791670955126)) + x);
	else
		tmp = Float64(x - Float64(y / fma(fma(fma(z, -0.18806319451591877, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -17000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.25e+14], N[(N[(N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-x) * y + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(N[(N[(z * -0.18806319451591877 + -0.5641895835477563), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7e7

   1. Initial program 89.0%

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

   3. Taylor expanded in x 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 -1.7e7 < z < 1.25e14

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto x + \left(\color{blue}{\frac{\frac{-5641895835477563}{5000000000000000} \cdot \left(y \cdot z\right)}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

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

   if 1.25e14 < z

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 96.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)\right) - \frac{5641895835477563}{5000000000000000}}}{y}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 80.7

          \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right)}, z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}{y}} \]

   7. Applied rewrites 80.7%

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)\right)}}{y}} \]

   8. Taylor expanded in x around 0

      \[\leadsto x + \frac{-1}{\frac{z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right) - \color{blue}{\frac{5641895835477563}{5000000000000000}}}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

       \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), \color{blue}{z}, -1.1283791670955126\right)}{y}} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. div-inv N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num-rev N/A

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

   12. Applied rewrites 80.8%

       \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, -0.18806319451591877, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{y - y \cdot z}{\mathsf{fma}\left(-x, y, 1.1283791670955126\right)} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, -0.18806319451591877, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 0.37:\\ \;\;\;\;\frac{-1}{x - \frac{1.1283791670955126}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -17000000.0)
   (+ (/ -1.0 x) x)
   (if (<= z 0.37)
     (+ (/ -1.0 (- x (/ 1.1283791670955126 y))) x)
     (- x (/ y (- (* y x) (* 1.1283791670955126 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -17000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 0.37) {
		tmp = (-1.0 / (x - (1.1283791670955126 / y))) + x;
	} else {
		tmp = x - (y / ((y * x) - (1.1283791670955126 * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-17000000.0d0)) then
        tmp = ((-1.0d0) / x) + x
    else if (z <= 0.37d0) then
        tmp = ((-1.0d0) / (x - (1.1283791670955126d0 / y))) + x
    else
        tmp = x - (y / ((y * x) - (1.1283791670955126d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -17000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 0.37) {
		tmp = (-1.0 / (x - (1.1283791670955126 / y))) + x;
	} else {
		tmp = x - (y / ((y * x) - (1.1283791670955126 * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -17000000.0:
		tmp = (-1.0 / x) + x
	elif z <= 0.37:
		tmp = (-1.0 / (x - (1.1283791670955126 / y))) + x
	else:
		tmp = x - (y / ((y * x) - (1.1283791670955126 * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 0.37)
		tmp = Float64(Float64(-1.0 / Float64(x - Float64(1.1283791670955126 / y))) + x);
	else
		tmp = Float64(x - Float64(y / Float64(Float64(y * x) - Float64(1.1283791670955126 * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -17000000.0)
		tmp = (-1.0 / x) + x;
	elseif (z <= 0.37)
		tmp = (-1.0 / (x - (1.1283791670955126 / y))) + x;
	else
		tmp = x - (y / ((y * x) - (1.1283791670955126 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -17000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 0.37], N[(N[(-1.0 / N[(x - N[(1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(N[(y * x), $MachinePrecision] - N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7e7

   1. Initial program 89.0%

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

   3. Taylor expanded in x 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 -1.7e7 < z < 0.37

   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

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\color{blue}{\left(x + \frac{-5641895835477563}{5000000000000000} \cdot \frac{z}{y}\right) - \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{y}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot \frac{1}{y}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

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

   if 0.37 < z

   1. Initial program 95.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 0.37:\\ \;\;\;\;\frac{-1}{x - \frac{1.1283791670955126}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - 1.1283791670955126 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - \mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -17000000.0)
   (+ (/ -1.0 x) x)
   (- x (/ y (- (* y x) (fma 1.1283791670955126 z 1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -17000000.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / ((y * x) - fma(1.1283791670955126, z, 1.1283791670955126)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / Float64(Float64(y * x) - fma(1.1283791670955126, z, 1.1283791670955126))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -17000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(N[(y * x), $MachinePrecision] - N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e7

   1. Initial program 89.0%

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

   3. Taylor expanded in x 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 -1.7e7 < z

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{y \cdot x - \mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -17000000.0)
   (+ (/ -1.0 x) x)
   (- x (/ y (fma x y -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -17000000.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -17000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e7

   1. Initial program 89.0%

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

   3. Taylor expanded in x 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 -1.7e7 < z

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{x \cdot y - \frac{5641895835477563}{5000000000000000}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ 0.8862269254527579 \cdot y + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* 0.8862269254527579 y) x))

C

double code(double x, double y, double z) {
	return (0.8862269254527579 * y) + 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 = (0.8862269254527579d0 * y) + x
end function

Java

public static double code(double x, double y, double z) {
	return (0.8862269254527579 * y) + x;
}

Python

def code(x, y, z):
	return (0.8862269254527579 * y) + x

Julia

function code(x, y, z)
	return Float64(Float64(0.8862269254527579 * y) + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.8862269254527579 * y) + x;
end

Wolfram

code[x_, y_, z_] := N[(N[(0.8862269254527579 * y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto x + \left(\color{blue}{\frac{\frac{-5641895835477563}{5000000000000000} \cdot \left(y \cdot z\right)}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 66.1%

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

6. Taylor expanded in x around 0

   \[\leadsto x + \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 46.7%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 53.1%

    \[\leadsto x + 0.8862269254527579 \cdot y \]

12. Final simplification 53.1%

    \[\leadsto 0.8862269254527579 \cdot y + x \]
13. 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 2024298 
(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)))))