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

Percentage Accurate: 95.4% → 98.4%

Time: 8.2s

Alternatives: 13

Speedup: 3.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, e^{z}, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x) 5e+286)
   (- x (/ y (fma -1.1283791670955126 (exp z) (* y x))))
   (+ (/ -1.0 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y / ((exp(z) * 1.1283791670955126) - (y * x))) + x) <= 5e+286) {
		tmp = x - (y / fma(-1.1283791670955126, exp(z), (y * x)));
	} else {
		tmp = (-1.0 / x) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x) <= 5e+286)
		tmp = Float64(x - Float64(y / fma(-1.1283791670955126, exp(z), Float64(y * x))));
	else
		tmp = Float64(Float64(-1.0 / x) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 5e+286], N[(x - N[(y / N[(-1.1283791670955126 * N[Exp[z], $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.0000000000000004e286

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if 5.0000000000000004e286 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 40.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, e^{z}, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_1 -1.0)
     t_0
     (if (<= t_1 0.05)
       (- x (/ y (fma -1.1283791670955126 z -1.1283791670955126)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = x - (y / fma(-1.1283791670955126, z, -1.1283791670955126));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = Float64(x - Float64(y / fma(-1.1283791670955126, z, -1.1283791670955126)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], t$95$0, If[LessEqual[t$95$1, 0.05], N[(x - N[(y / N[(-1.1283791670955126 * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $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)))) < -1 or 0.050000000000000003 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -1 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.050000000000000003

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 94.1

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 81.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq -1:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 0.05:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{0.8862269254527579}{1 + z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_1 -1.0)
     t_0
     (if (<= t_1 0.05) (* (/ 0.8862269254527579 (+ 1.0 z)) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((exp(z) * 1.1283791670955126d0) - (y * x))) + x
    if (t_1 <= (-1.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.05d0) then
        tmp = (0.8862269254527579d0 / (1.0d0 + z)) * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((Math.exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((math.exp(z) * 1.1283791670955126) - (y * x))) + x
	tmp = 0
	if t_1 <= -1.0:
		tmp = t_0
	elif t_1 <= 0.05:
		tmp = (0.8862269254527579 / (1.0 + z)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(0.8862269254527579 / Float64(1.0 + z)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = (0.8862269254527579 / (1.0 + z)) * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], t$95$0, If[LessEqual[t$95$1, 0.05], N[(N[(0.8862269254527579 / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * y), $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)))) < -1 or 0.050000000000000003 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -1 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.050000000000000003

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 26.6

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   7. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 26.7%

       \[\leadsto \frac{y}{1 + z} \cdot 0.8862269254527579 \]
   11. Step-by-step derivation

   12. Applied rewrites 26.7%

       \[\leadsto y \cdot \color{blue}{\frac{0.8862269254527579}{1 + z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq -1:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 0.05:\\ \;\;\;\;\frac{0.8862269254527579}{1 + z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* (exp z) 1.1283791670955126) (* y x))) x)))
   (if (<= t_1 -1.0) t_0 (if (<= t_1 0.05) (* 0.8862269254527579 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = 0.8862269254527579 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) + x
    t_1 = (y / ((exp(z) * 1.1283791670955126d0) - (y * x))) + x
    if (t_1 <= (-1.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.05d0) then
        tmp = 0.8862269254527579d0 * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((Math.exp(z) * 1.1283791670955126) - (y * x))) + x;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0;
	} else if (t_1 <= 0.05) {
		tmp = 0.8862269254527579 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 / x) + x
	t_1 = (y / ((math.exp(z) * 1.1283791670955126) - (y * x))) + x
	tmp = 0
	if t_1 <= -1.0:
		tmp = t_0
	elif t_1 <= 0.05:
		tmp = 0.8862269254527579 * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = Float64(0.8862269254527579 * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 / x) + x;
	t_1 = (y / ((exp(z) * 1.1283791670955126) - (y * x))) + x;
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = t_0;
	elseif (t_1 <= 0.05)
		tmp = 0.8862269254527579 * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], t$95$0, If[LessEqual[t$95$1, 0.05], N[(0.8862269254527579 * y), $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)))) < -1 or 0.050000000000000003 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -1 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.050000000000000003

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 26.6

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   7. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 26.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq -1:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{e^{z} \cdot 1.1283791670955126 - y \cdot x} + x \leq 0.05:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.0%

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

4. Final simplification 98.4%

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

Alternative 6: 96.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 98.0%

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

Alternative 7: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -118:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, \mathsf{fma}\left(y, x, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -118.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.35e+19)
     (- x (/ y (fma -1.1283791670955126 z (fma y x -1.1283791670955126))))
     (-
      x
      (/
       y
       (fma
        (fma
         (fma -0.18806319451591877 z -0.5641895835477563)
         z
         -1.1283791670955126)
        z
        -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -118.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.35e+19) {
		tmp = x - (y / fma(-1.1283791670955126, z, fma(y, x, -1.1283791670955126)));
	} else {
		tmp = x - (y / fma(fma(fma(-0.18806319451591877, z, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -118.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.35e+19)
		tmp = Float64(x - Float64(y / fma(-1.1283791670955126, z, fma(y, x, -1.1283791670955126))));
	else
		tmp = Float64(x - Float64(y / fma(fma(fma(-0.18806319451591877, z, -0.5641895835477563), z, -1.1283791670955126), z, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -118.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+19], N[(x - N[(y / N[(-1.1283791670955126 * z + N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(N[(-0.18806319451591877 * z + -0.5641895835477563), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -118

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -118 < z < 1.35e19

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.35e19 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -118:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, \mathsf{fma}\left(y, x, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -118:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, \mathsf{fma}\left(y, x, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -118.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.35e+19)
     (- x (/ y (fma -1.1283791670955126 z (fma y x -1.1283791670955126))))
     (-
      x
      (/
       y
       (fma
        (fma -0.5641895835477563 z -1.1283791670955126)
        z
        -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -118.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.35e+19) {
		tmp = x - (y / fma(-1.1283791670955126, z, fma(y, x, -1.1283791670955126)));
	} else {
		tmp = x - (y / fma(fma(-0.5641895835477563, z, -1.1283791670955126), z, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -118.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.35e+19)
		tmp = Float64(x - Float64(y / fma(-1.1283791670955126, z, fma(y, x, -1.1283791670955126))));
	else
		tmp = Float64(x - Float64(y / fma(fma(-0.5641895835477563, z, -1.1283791670955126), z, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -118.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+19], N[(x - N[(y / N[(-1.1283791670955126 * z + N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(-0.5641895835477563 * z + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -118

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -118 < z < 1.35e19

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.35e19 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 95.2

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -118:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, \mathsf{fma}\left(y, x, -1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -185.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.35e+19)
     (- x (/ y (fma y x -1.1283791670955126)))
     (-
      x
      (/
       y
       (fma
        (fma -0.5641895835477563 z -1.1283791670955126)
        z
        -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -185.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.35e+19) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = x - (y / fma(fma(-0.5641895835477563, z, -1.1283791670955126), z, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -185.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.35e+19)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / fma(fma(-0.5641895835477563, z, -1.1283791670955126), z, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -185.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+19], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(-0.5641895835477563 * z + -1.1283791670955126), $MachinePrecision] * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -185

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -185 < z < 1.35e19

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.35e19 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 95.2

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5641895835477563, z, -1.1283791670955126\right), z, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -185.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.35e+19)
     (- x (/ y (fma y x -1.1283791670955126)))
     (- x (/ y (* (* z z) -0.5641895835477563))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -185.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.35e+19) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = x - (y / ((z * z) * -0.5641895835477563));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -185.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.35e+19)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(z * z) * -0.5641895835477563)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -185.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+19], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(z * z), $MachinePrecision] * -0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -185

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -185 < z < 1.35e19

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.35e19 < z

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 95.2

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 95.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot z\right) \cdot -0.5641895835477563}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+129}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -185.0)
   (+ (/ -1.0 x) x)
   (if (<= z 6.4e+129)
     (- x (/ y (fma y x -1.1283791670955126)))
     (- x (/ y (fma -1.1283791670955126 z -1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -185.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 6.4e+129) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = x - (y / fma(-1.1283791670955126, z, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -185.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 6.4e+129)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = Float64(x - Float64(y / fma(-1.1283791670955126, z, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -185.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.4e+129], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(-1.1283791670955126 * z + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -185

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -185 < z < 6.4000000000000005e129

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 95.2

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

   7. Applied rewrites 95.2%

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

   if 6.4000000000000005e129 < z

   1. Initial program 97.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 83.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+129}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(-1.1283791670955126, z, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -45:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -45.0) (/ -1.0 x) (* 0.8862269254527579 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -45.0) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.8862269254527579 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-45.0d0)) then
        tmp = (-1.0d0) / x
    else
        tmp = 0.8862269254527579d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -45.0) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.8862269254527579 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -45.0:
		tmp = -1.0 / x
	else:
		tmp = 0.8862269254527579 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -45.0)
		tmp = Float64(-1.0 / x);
	else
		tmp = Float64(0.8862269254527579 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -45.0)
		tmp = -1.0 / x;
	else
		tmp = 0.8862269254527579 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -45.0], N[(-1.0 / x), $MachinePrecision], N[(0.8862269254527579 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -45

   1. Initial program 87.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 48.1%

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

   if -45 < z

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 18.7

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   7. Applied rewrites 18.7%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 18.4%

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

Alternative 13: 14.4% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. unsub-neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. distribute-neg-in N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. remove-double-neg N/A

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

   18. lower-fma.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-exp.f64 15.6

      \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

7. Applied rewrites 15.6%

   \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

8. Taylor expanded in z around 0

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

10. Applied rewrites 15.4%

    \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
11. 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 2024294 
(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)))))