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

Percentage Accurate: 95.5% → 99.9%

Time: 11.5s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto x + \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -200.0)
     t_0
     (if (<= t_1 4e-8)
       (fma y (fma y (* x 0.7853981633974483) 0.8862269254527579) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = fma(y, fma(y, (x * 0.7853981633974483), 0.8862269254527579), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = fma(y, fma(y, Float64(x * 0.7853981633974483), 0.8862269254527579), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$0, If[LessEqual[t$95$1, 4e-8], N[(y * N[(y * N[(x * 0.7853981633974483), $MachinePrecision] + 0.8862269254527579), $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)))) < -200 or 4.0000000000000001e-8 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -200 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4.0000000000000001e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   7. 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. metadata-eval N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 57.8

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

   8. Applied rewrites 57.8%

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

   9. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(1 + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot {y}^{2}\right) - \color{blue}{\frac{-5000000000000000}{5641895835477563} \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.0%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, 0.7853981633974483 \cdot x, 0.8862269254527579\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -200.0)
     t_0
     (if (<= t_1 4e-8) (fma y 0.8862269254527579 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$0, If[LessEqual[t$95$1, 4e-8], N[(y * 0.8862269254527579 + 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)))) < -200 or 4.0000000000000001e-8 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -200 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4.0000000000000001e-8

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. rec-exp N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

4. Final simplification 98.8%

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

Alternative 6: 97.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 97.8

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

   7. Applied rewrites 97.8%

      \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.18806319451591877, -0.5641895835477563\right), -1.1283791670955126\right), -1.1283791670955126\right)}\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 97.8

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in z around inf

      \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, \frac{-5641895835477563}{30000000000000000} \cdot \color{blue}{{z}^{2}}, \frac{-5641895835477563}{5000000000000000}\right)\right)}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

       \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, -0.18806319451591877 \cdot \color{blue}{\left(z \cdot z\right)}, -1.1283791670955126\right)\right)}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 9: 96.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.7%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000} \cdot \color{blue}{{z}^{2}}, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.5%

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

4. Final simplification 97.5%

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

Alternative 10: 95.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 11: 92.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 91.7

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

   5. Applied rewrites 91.7%

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

Alternative 12: 89.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

      \[\leadsto x + \frac{-1}{\color{blue}{x + \frac{-1.1283791670955126}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   7. 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. metadata-eval N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 14: 61.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.2e+26) (/ -1.0 x) (- x (/ y -1.1283791670955126))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e+26) {
		tmp = -1.0 / x;
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	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 <= (-4.2d+26)) then
        tmp = (-1.0d0) / x
    else
        tmp = x - (y / (-1.1283791670955126d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e+26) {
		tmp = -1.0 / x;
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.2e+26:
		tmp = -1.0 / x
	else:
		tmp = x - (y / -1.1283791670955126)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.2e+26)
		tmp = Float64(-1.0 / x);
	else
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.2e+26)
		tmp = -1.0 / x;
	else
		tmp = x - (y / -1.1283791670955126);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.2e+26], N[(-1.0 / x), $MachinePrecision], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e26

   1. Initial program 90.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.7%

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

   if -4.2000000000000002e26 < z

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   7. 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. metadata-eval N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 87.8

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

   8. Applied rewrites 87.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 66.8%

       \[\leadsto x - \frac{y}{-1.1283791670955126} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.2e+26) (/ -1.0 x) (fma y 0.8862269254527579 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e+26) {
		tmp = -1.0 / x;
	} else {
		tmp = fma(y, 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.2e+26)
		tmp = Float64(-1.0 / x);
	else
		tmp = fma(y, 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.2e+26], N[(-1.0 / x), $MachinePrecision], N[(y * 0.8862269254527579 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e26

   1. Initial program 90.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.7%

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

   if -4.2000000000000002e26 < z

   1. Initial program 97.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. rec-exp N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.7%

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

Alternative 16: 59.6% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 0.8862269254527579, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y 0.8862269254527579 x))

C

double code(double x, double y, double z) {
	return fma(y, 0.8862269254527579, x);
}

Julia

function code(x, y, z)
	return fma(y, 0.8862269254527579, x)
end

Wolfram

code[x_, y_, z_] := N[(y * 0.8862269254527579 + 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 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. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. rec-exp N/A

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

   9. neg-mul-1 N/A

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

   10. lower-exp.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 56.6%

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

Alternative 17: 14.1% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * 0.8862269254527579d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * 0.8862269254527579), $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 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. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. rec-exp N/A

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

   9. neg-mul-1 N/A

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

   10. lower-exp.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 56.6%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 14.2%

    \[\leadsto 0.8862269254527579 \cdot y \]

12. Final simplification 14.2%

    \[\leadsto y \cdot 0.8862269254527579 \]
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 2024233 
(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)))))