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

Percentage Accurate: 95.5% → 99.9%

Time: 12.7s

Alternatives: 15

Speedup: 3.2×

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, 0.4× speedup

Math

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

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)))
    (pow (/ 1.0 y) -1.0)
    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))), pow((1.0 / y), -1.0), 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))), (Float64(1.0 / y) ^ -1.0), 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] * N[Power[N[(1.0 / y), $MachinePrecision], -1.0], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lift--.f64 N/A

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

      14. flip-- N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 99.8%

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

Alternative 2: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{x \cdot x}{x}\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -50000.0)
     t_0
     (if (<= t_1 -2e-153)
       (/ (* x x) x)
       (if (<= t_1 0.1) (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 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= -2e-153) {
		tmp = (x * x) / x;
	} else if (t_1 <= 0.1) {
		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 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= -2e-153)
		tmp = Float64(Float64(x * x) / x);
	elseif (t_1 <= 0.1)
		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, -50000.0], t$95$0, If[LessEqual[t$95$1, -2e-153], N[(N[(x * x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(y * 0.8862269254527579 + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -5e4 or 0.10000000000000001 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 92.6

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

   5. Simplified 92.6%

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

   if -5e4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -2.00000000000000008e-153

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 1.6

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

   5. Simplified 1.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{2} - 1}{x}} \]

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 1.6

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

   8. Simplified 1.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. lower-*.f64 84.0

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

   11. Simplified 84.0%

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

   if -2.00000000000000008e-153 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 75.4

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

   5. Simplified 75.4%

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

   7. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y \cdot \left(x \cdot \left(y \cdot x\right)\right) - 1.2732395447351628 \cdot \left(\left(1 + z\right) \cdot \left(1 + z\right)\right)}, x \cdot \left(-y\right) - \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right), x\right)} \]

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-1 \cdot \left(x \cdot y\right) - \frac{5641895835477563}{5000000000000000}}{{x}^{2} \cdot {y}^{2} - \frac{31830988618379068626528276418969}{25000000000000000000000000000000}}, x\right)} \]

   10. Simplified 72.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(x, -y, -1.1283791670955126\right)}{\mathsf{fma}\left(x \cdot y, x \cdot y, -1.2732395447351628\right)}, x\right)} \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 68.3

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

   13. Simplified 68.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{x \cdot x}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -50000.0)
     t_0
     (if (<= t_1 0.1) (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 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		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 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		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, -50000.0], t$95$0, If[LessEqual[t$95$1, 0.1], 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)))) < -5e4 or 0.10000000000000001 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 92.6

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

   5. Simplified 92.6%

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

   if -5e4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 78.7

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

   5. Simplified 78.7%

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

   7. Applied egg-rr 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y \cdot \left(x \cdot \left(y \cdot x\right)\right) - 1.2732395447351628 \cdot \left(\left(1 + z\right) \cdot \left(1 + z\right)\right)}, x \cdot \left(-y\right) - \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right), x\right)} \]

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-1 \cdot \left(x \cdot y\right) - \frac{5641895835477563}{5000000000000000}}{{x}^{2} \cdot {y}^{2} - \frac{31830988618379068626528276418969}{25000000000000000000000000000000}}, x\right)} \]

   10. Simplified 72.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(x, -y, -1.1283791670955126\right)}{\mathsf{fma}\left(x \cdot y, x \cdot y, -1.2732395447351628\right)}, x\right)} \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 67.7

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

   13. Simplified 67.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+200}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 2e+200) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 2e+200) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 2d+200) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 2e+200) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 2e+200:
		tmp = t_0
	else:
		tmp = x + (-1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 2e+200)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 2e+200)
		tmp = t_0;
	else
		tmp = x + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+200], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1.9999999999999999e200

   1. Initial program 99.1%

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

   if 1.9999999999999999e200 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 72.9%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 2 \cdot 10^{+200}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (*
      x
      (-
       (/
        (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 / (x * ((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(x * Float64(Float64(fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), 1.1283791670955126) / 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[(x * N[(N[(N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 98.1

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

   8. Simplified 98.1%

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

4. Final simplification 98.4%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (*
      x
      (-
       (/
        (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 / (x * ((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(x * Float64(Float64(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) / 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[(x * N[(N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.3%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 98.1

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

   8. Simplified 98.1%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 97.2

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

   11. Simplified 97.2%

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

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. remove-double-neg N/A

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

       7. lift-neg.f64 N/A

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

       8. distribute-frac-neg2 N/A

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

       9. lift-neg.f64 N/A

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

       10. remove-double-neg N/A

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

   13. Applied egg-rr 97.2%

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

4. Final simplification 97.7%

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

Alternative 7: 96.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1700000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), \mathsf{fma}\left(y, -x, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 1700000000.0)
     (+
      x
      (/
       y
       (fma
        z
        (fma
         z
         (fma z 0.18806319451591877 0.5641895835477563)
         1.1283791670955126)
        (fma y (- x) 1.1283791670955126))))
     (fma 5.317361552716548 (/ y (* z (* z z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1700000000.0) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), fma(y, -x, 1.1283791670955126)));
	} else {
		tmp = fma(5.317361552716548, (y / (z * (z * z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1700000000.0)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), fma(y, Float64(-x), 1.1283791670955126))));
	else
		tmp = fma(5.317361552716548, Float64(y / Float64(z * Float64(z * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1700000000.0], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.317361552716548 * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 1.7e9

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.3

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

   5. Simplified 99.3%

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

   if 1.7e9 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{{z}^{3}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{3}}}, x\right) \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{z \cdot \color{blue}{{z}^{2}}}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{\color{blue}{z \cdot {z}^{2}}}, x\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 90.0

         \[\leadsto \mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \color{blue}{\left(z \cdot z\right)}}, x\right) \]

   8. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1700000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 1700000000.0)
     (+
      x
      (/
       y
       (-
        (fma
         z
         (fma z 0.5641895835477563 1.1283791670955126)
         1.1283791670955126)
        (* x y))))
     (fma 5.317361552716548 (/ y (* z (* z z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1700000000.0) {
		tmp = x + (y / (fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma(5.317361552716548, (y / (z * (z * z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1700000000.0)
		tmp = Float64(x + Float64(y / Float64(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(5.317361552716548, Float64(y / Float64(z * Float64(z * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1700000000.0], N[(x + N[(y / N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.317361552716548 * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 1.7e9

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.3

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

   5. Simplified 99.3%

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

   if 1.7e9 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{{z}^{3}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{3}}}, x\right) \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{z \cdot \color{blue}{{z}^{2}}}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{\color{blue}{z \cdot {z}^{2}}}, x\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 90.0

         \[\leadsto \mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \color{blue}{\left(z \cdot z\right)}}, x\right) \]

   8. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1700000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1600000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 1600000000.0)
     (+ x (/ y (fma y (- x) (fma z 1.1283791670955126 1.1283791670955126))))
     (fma 5.317361552716548 (/ y (* z (* z z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1600000000.0) {
		tmp = x + (y / fma(y, -x, fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = fma(5.317361552716548, (y / (z * (z * z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1600000000.0)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = fma(5.317361552716548, Float64(y / Float64(z * Float64(z * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1600000000.0], N[(x + N[(y / N[(y * (-x) + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.317361552716548 * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 1.6e9

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if 1.6e9 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{{z}^{3}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{3}}}, x\right) \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{z \cdot \color{blue}{{z}^{2}}}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{\color{blue}{z \cdot {z}^{2}}}, x\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 90.0

         \[\leadsto \mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \color{blue}{\left(z \cdot z\right)}}, x\right) \]

   8. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 96.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1600000000:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 1600000000.0)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma 5.317361552716548 (/ y (* z (* z z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1600000000.0) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma(5.317361552716548, (y / (z * (z * z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1600000000.0)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(5.317361552716548, Float64(y / Float64(z * Float64(z * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1600000000.0], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.317361552716548 * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 1.6e9

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lift--.f64 N/A

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

      14. flip-- N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 98.9

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

   7. Simplified 98.9%

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

   if 1.6e9 < z

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 82.9

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

   5. Simplified 82.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{{z}^{3}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{3}}}, x\right) \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{z \cdot \color{blue}{{z}^{2}}}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{\color{blue}{z \cdot {z}^{2}}}, x\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 90.0

         \[\leadsto \mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \color{blue}{\left(z \cdot z\right)}}, x\right) \]

   8. Simplified 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 95.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1.7724538509055159}{z \cdot z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 2.5e+40)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma y (/ 1.7724538509055159 (* z z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.5e+40) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma(y, (1.7724538509055159 / (z * z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 2.5e+40)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(y, Float64(1.7724538509055159 / Float64(z * z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+40], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.7724538509055159 / N[(z * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 2.50000000000000002e40

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lift--.f64 N/A

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

      14. flip-- N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 96.6

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

   7. Simplified 96.6%

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

   if 2.50000000000000002e40 < z

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 96.0

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

   11. Simplified 96.0%

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

   12. Taylor expanded in z around inf

       \[\leadsto \color{blue}{x + \frac{10000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{2}}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{10000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{2}} + x} \]

       2. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{10000000000000000}{5641895835477563} \cdot y}{{z}^{2}}} + x \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot \frac{10000000000000000}{5641895835477563}}}{{z}^{2}} + x \]

       4. associate-/l* N/A

          \[\leadsto \color{blue}{y \cdot \frac{\frac{10000000000000000}{5641895835477563}}{{z}^{2}}} + x \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\frac{10000000000000000}{5641895835477563}}{{z}^{2}}, x\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{10000000000000000}{5641895835477563}}{{z}^{2}}}, x\right) \]

       7. unpow2 N/A

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

       8. lower-*.f64 91.9

          \[\leadsto \mathsf{fma}\left(y, \frac{1.7724538509055159}{\color{blue}{z \cdot z}}, x\right) \]

   14. Simplified 91.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1.7724538509055159}{z \cdot z}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+18)
   (+ x (/ -1.0 x))
   (if (<= z 3.1e+40)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma 0.8862269254527579 (/ y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+18) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.1e+40) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma(0.8862269254527579, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+18)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.1e+40)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(0.8862269254527579, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+18], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+40], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8862269254527579 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1e18

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 99.9

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

   5. Simplified 99.9%

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

   if -5.1e18 < z < 3.0999999999999998e40

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. inv-pow N/A

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

      10. div-inv N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. lift--.f64 N/A

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

      14. flip-- N/A

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

      15. clear-num N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 96.6

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

   7. Simplified 96.6%

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

   if 3.0999999999999998e40 < z

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 77.2

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

   5. Simplified 77.2%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 76.7

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

   8. Simplified 76.7%

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

Alternative 13: 60.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e+105) (/ -1.0 x) (fma y 0.8862269254527579 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+105) {
		tmp = -1.0 / x;
	} else {
		tmp = fma(y, 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e+105)
		tmp = Float64(-1.0 / x);
	else
		tmp = fma(y, 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e+105], N[(-1.0 / x), $MachinePrecision], N[(y * 0.8862269254527579 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e105

   1. Initial program 92.6%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 46.0

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

   8. Simplified 46.0%

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

   if -1.6e105 < y

   1. Initial program 96.8%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 88.5

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

   5. Simplified 88.5%

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

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y \cdot \left(x \cdot \left(y \cdot x\right)\right) - 1.2732395447351628 \cdot \left(\left(1 + z\right) \cdot \left(1 + z\right)\right)}, x \cdot \left(-y\right) - \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right), x\right)} \]

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-1 \cdot \left(x \cdot y\right) - \frac{5641895835477563}{5000000000000000}}{{x}^{2} \cdot {y}^{2} - \frac{31830988618379068626528276418969}{25000000000000000000000000000000}}, x\right)} \]

   10. Simplified 74.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(x, -y, -1.1283791670955126\right)}{\mathsf{fma}\left(x \cdot y, x \cdot y, -1.2732395447351628\right)}, x\right)} \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 73.1

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

   13. Simplified 73.1%

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

Alternative 14: 59.7% 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 96.1%

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

3. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 86.6

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

5. Simplified 86.6%

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

7. Applied egg-rr 69.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y \cdot \left(x \cdot \left(y \cdot x\right)\right) - 1.2732395447351628 \cdot \left(\left(1 + z\right) \cdot \left(1 + z\right)\right)}, x \cdot \left(-y\right) - \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right), x\right)} \]

8. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-1 \cdot \left(x \cdot y\right) - \frac{5641895835477563}{5000000000000000}}{{x}^{2} \cdot {y}^{2} - \frac{31830988618379068626528276418969}{25000000000000000000000000000000}}, x\right)} \]

10. Simplified 71.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(x, -y, -1.1283791670955126\right)}{\mathsf{fma}\left(x \cdot y, x \cdot y, -1.2732395447351628\right)}, x\right)} \]

11. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 64.0

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

13. Simplified 64.0%

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

Alternative 15: 14.2% 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 96.1%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. mul-1-neg N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 85.8

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

5. Simplified 85.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 14.6

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

8. Simplified 14.6%

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

9. Taylor expanded in z around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 14.6

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

11. Simplified 14.6%

    \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
12. 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 2024207 
(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)))))