Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 11.4s

Alternatives: 22

Speedup: 1.0×

• 59.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -50.0)
     (exp x)
     (if (<= t_0 -1e-147)
       (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
       (if (<= t_0 50000.0)
         (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
         (pow y y))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = exp(x);
	} else if (t_0 <= -1e-147) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else if (t_0 <= 50000.0) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = exp(x);
	elseif (t_0 <= -1e-147)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	elseif (t_0 <= 50000.0)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = y ^ y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, -1e-147], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -50

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 98.7

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.4

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

   8. Simplified 89.4%

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

   if -50 < (+.f64 x (*.f64 y (log.f64 y))) < -9.9999999999999997e-148

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 95.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 95.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 56.5

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 56.5

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 81.8

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 81.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if -9.9999999999999997e-148 < (+.f64 x (*.f64 y (log.f64 y))) < 5e4

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 100.0

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. lower-fma.f64 74.9

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if 5e4 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. lower-log.f64 76.5

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 76.5%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log y} \cdot y} \]

      4. pow-to-exp N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      5. lift-pow.f64 76.5

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 76.5%

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

Alternative 3: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\frac{2 + \frac{4 + \frac{4 + \frac{-8}{z \cdot z}}{z}}{z}}{z \cdot z}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)))
   (if (<= t_0 -4e+107)
     (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
     (if (<= t_0 -5e+21)
       (/ 1.0 (/ (+ 2.0 (/ (+ 4.0 (/ (+ 4.0 (/ -8.0 (* z z))) z)) z)) (* z z)))
       (if (<= t_0 2e+139)
         (/
          (fma z (* z (* (fma z 0.5 -1.0) (fma z 0.5 -1.0))) -1.0)
          (fma z (fma z 0.5 -1.0) -1.0))
         (fma (* z (fma (* z z) 0.25 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double tmp;
	if (t_0 <= -4e+107) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else if (t_0 <= -5e+21) {
		tmp = 1.0 / ((2.0 + ((4.0 + ((4.0 + (-8.0 / (z * z))) / z)) / z)) / (z * z));
	} else if (t_0 <= 2e+139) {
		tmp = fma(z, (z * (fma(z, 0.5, -1.0) * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else {
		tmp = fma((z * fma((z * z), 0.25, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_0 <= -4e+107)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	elseif (t_0 <= -5e+21)
		tmp = Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(4.0 + Float64(Float64(4.0 + Float64(-8.0 / Float64(z * z))) / z)) / z)) / Float64(z * z)));
	elseif (t_0 <= 2e+139)
		tmp = Float64(fma(z, Float64(z * Float64(fma(z, 0.5, -1.0) * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0));
	else
		tmp = fma(Float64(z * fma(Float64(z * z), 0.25, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+107], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e+21], N[(1.0 / N[(N[(2.0 + N[(N[(4.0 + N[(N[(4.0 + N[(-8.0 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+139], N[(N[(z * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(z * z), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -3.9999999999999999e107

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 64.9

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 64.9%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 2.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 2.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 2.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 2.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 68.5

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 68.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if -3.9999999999999999e107 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -5e21

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 31.4

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 31.4%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 2.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 2.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 2.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 2.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around -inf

       \[\leadsto \frac{1}{\color{blue}{\frac{2 + -1 \cdot \frac{-1 \cdot \frac{4 - 8 \cdot \frac{1}{{z}^{2}}}{z} - 4}{z}}{{z}^{2}}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{2 + -1 \cdot \frac{-1 \cdot \frac{4 - 8 \cdot \frac{1}{{z}^{2}}}{z} - 4}{z}}{{z}^{2}}}} \]

   13. Simplified 37.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{2 + \frac{4 + \frac{4 + \frac{-8}{z \cdot z}}{z}}{z}}{z \cdot z}}} \]

   if -5e21 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2.00000000000000007e139

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 49.0

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 49.0%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 38.0

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. swap-sqr N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{z \cdot \left(\left(z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right) \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right) + \color{blue}{-1}}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) - 1} \]

      10. lower-fma.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right), -1\right)}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + \color{blue}{-1}} \]

      16. lower-fma.f64 46.6

          \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}} \]

   10. Applied egg-rr 46.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}} \]

   if 2.00000000000000007e139 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 41.2

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 41.2%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 44.7

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot z} + 1 \]

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied egg-rr 49.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -50.0) (exp x) (if (<= t_0 50000.0) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = exp(x);
	} else if (t_0 <= 50000.0) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, 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) :: t_0
    real(8) :: tmp
    t_0 = x + (y * log(y))
    if (t_0 <= (-50.0d0)) then
        tmp = exp(x)
    else if (t_0 <= 50000.0d0) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y * Math.log(y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = Math.exp(x);
	} else if (t_0 <= 50000.0) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y * math.log(y))
	tmp = 0
	if t_0 <= -50.0:
		tmp = math.exp(x)
	elif t_0 <= 50000.0:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = exp(x);
	elseif (t_0 <= 50000.0)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y * log(y));
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = exp(x);
	elseif (t_0 <= 50000.0)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -50

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 98.7

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.4

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

   8. Simplified 89.4%

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

   if -50 < (+.f64 x (*.f64 y (log.f64 y))) < 5e4

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 97.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 97.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   if 5e4 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. lower-log.f64 76.5

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 76.5%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log y} \cdot y} \]

      4. pow-to-exp N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      5. lift-pow.f64 76.5

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 76.5%

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

Alternative 5: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq -5 \cdot 10^{-303}:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) -5e-303)
   (* (pow y y) (exp (- x z)))
   (exp (fma y (log y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= -5e-303) {
		tmp = pow(y, y) * exp((x - z));
	} else {
		tmp = exp(fma(y, log(y), x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= -5e-303)
		tmp = Float64((y ^ y) * exp(Float64(x - z)));
	else
		tmp = exp(fma(y, log(y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], -5e-303], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < -4.9999999999999998e-303

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. exp-sum N/A

         \[\leadsto \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]

      7. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \cdot e^{x - z} \]

      8. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \cdot e^{x - z} \]

      9. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log y} \cdot y} \cdot e^{x - z} \]

      10. exp-to-pow N/A

          \[\leadsto \color{blue}{{y}^{y}} \cdot e^{x - z} \]

      11. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{y}^{y}} \cdot e^{x - z} \]

      12. lower-exp.f64 N/A

          \[\leadsto {y}^{y} \cdot \color{blue}{e^{x - z}} \]

      13. lower--.f64 100.0

          \[\leadsto {y}^{y} \cdot e^{\color{blue}{x - z}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]

   if -4.9999999999999998e-303 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x + y \cdot \log y}} \]

      2. +-commutative N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y + x}} \]

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 90.9

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

   5. Simplified 90.9%

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

Alternative 6: 50.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)))
   (if (<= t_0 -2e+34)
     (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
     (if (<= t_0 2e+139)
       (/
        (fma z (* z (* (fma z 0.5 -1.0) (fma z 0.5 -1.0))) -1.0)
        (fma z (fma z 0.5 -1.0) -1.0))
       (fma (* z (fma (* z z) 0.25 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double tmp;
	if (t_0 <= -2e+34) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e+139) {
		tmp = fma(z, (z * (fma(z, 0.5, -1.0) * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else {
		tmp = fma((z * fma((z * z), 0.25, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_0 <= -2e+34)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e+139)
		tmp = Float64(fma(z, Float64(z * Float64(fma(z, 0.5, -1.0) * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0));
	else
		tmp = fma(Float64(z * fma(Float64(z * z), 0.25, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+34], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+139], N[(N[(z * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(z * z), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1.99999999999999989e34

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 60.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 60.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 2.2

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 2.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 2.2

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 2.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 58.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 58.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if -1.99999999999999989e34 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2.00000000000000007e139

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 46.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 46.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 36.3

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. swap-sqr N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{z \cdot \left(\left(z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right) \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right) + \color{blue}{-1}}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) - 1} \]

      10. lower-fma.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right)\right), -1\right)}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + \color{blue}{-1}} \]

      16. lower-fma.f64 44.4

          \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}} \]

   10. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}} \]

   if 2.00000000000000007e139 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 41.2

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 41.2%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 44.7

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot z} + 1 \]

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied egg-rr 49.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 49.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)))
   (if (<= t_0 2e-12)
     (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
     (if (<= t_0 2e+139)
       (/ (fma 0.25 (* (* z z) (* z z)) -1.0) (fma z (* z 0.5) -1.0))
       (fma (* z (fma (* z z) 0.25 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double tmp;
	if (t_0 <= 2e-12) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e+139) {
		tmp = fma(0.25, ((z * z) * (z * z)), -1.0) / fma(z, (z * 0.5), -1.0);
	} else {
		tmp = fma((z * fma((z * z), 0.25, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_0 <= 2e-12)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e+139)
		tmp = Float64(fma(0.25, Float64(Float64(z * z) * Float64(z * z)), -1.0) / fma(z, Float64(z * 0.5), -1.0));
	else
		tmp = fma(Float64(z * fma(Float64(z * z), 0.25, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-12], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+139], N[(N[(0.25 * N[(N[(z * z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(z * N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(z * z), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.99999999999999996e-12

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 66.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 66.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 26.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 26.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 26.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 26.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 65.2

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 65.2%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if 1.99999999999999996e-12 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2.00000000000000007e139

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 22.0

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 22.0%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 3.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 3.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}}, 1\right) \]

       2. lower-*.f64 3.6

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]

   11. Simplified 3.6%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. sub-neg N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. unpow-prod-down N/A

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

       12. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{\frac{1}{4}} \cdot {\left(z \cdot z\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       13. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{4} \cdot {\left(z \cdot z\right)}^{2} + \color{blue}{-1}}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       14. lower-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(z \cdot z\right)}^{2}, -1\right)}}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       15. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(z \cdot z\right) \cdot \left(z \cdot z\right)}, -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(z \cdot z\right) \cdot \left(z \cdot z\right)}, -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       17. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\color{blue}{z \cdot \left(z \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

       18. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) + \color{blue}{-1}} \]

       19. lower-fma.f64 18.0

           \[\leadsto \frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\color{blue}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}} \]

   13. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}} \]

   if 2.00000000000000007e139 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 41.2

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 41.2%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 44.7

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot z} + 1 \]

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied egg-rr 49.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 32.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := z \cdot \left(z \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -200:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1.6 \cdot 10^{+108}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* z (* z 0.5))))
   (if (<= t_0 -200.0) t_1 (if (<= t_0 1.6e+108) (- 1.0 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = z * (z * 0.5);
	double tmp;
	if (t_0 <= -200.0) {
		tmp = t_1;
	} else if (t_0 <= 1.6e+108) {
		tmp = 1.0 - z;
	} else {
		tmp = t_1;
	}
	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 = (x + (y * log(y))) - z
    t_1 = z * (z * 0.5d0)
    if (t_0 <= (-200.0d0)) then
        tmp = t_1
    else if (t_0 <= 1.6d+108) then
        tmp = 1.0d0 - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + (y * Math.log(y))) - z;
	double t_1 = z * (z * 0.5);
	double tmp;
	if (t_0 <= -200.0) {
		tmp = t_1;
	} else if (t_0 <= 1.6e+108) {
		tmp = 1.0 - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + (y * math.log(y))) - z
	t_1 = z * (z * 0.5)
	tmp = 0
	if t_0 <= -200.0:
		tmp = t_1
	elif t_0 <= 1.6e+108:
		tmp = 1.0 - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(z * Float64(z * 0.5))
	tmp = 0.0
	if (t_0 <= -200.0)
		tmp = t_1;
	elseif (t_0 <= 1.6e+108)
		tmp = Float64(1.0 - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + (y * log(y))) - z;
	t_1 = z * (z * 0.5);
	tmp = 0.0;
	if (t_0 <= -200.0)
		tmp = t_1;
	elseif (t_0 <= 1.6e+108)
		tmp = 1.0 - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200.0], t$95$1, If[LessEqual[t$95$0, 1.6e+108], N[(1.0 - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -200 or 1.6e108 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 47.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 47.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 24.8

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 31.2

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 31.2%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -200 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.6e108

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 51.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 51.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - z} \]

      3. lower--.f64 44.1

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 44.1%

      \[\leadsto \color{blue}{1 - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+77}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;z \leq 10^{+19}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{z - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+77)
   (exp (- x z))
   (if (<= z 1e+19) (exp (fma y (log y) x)) (/ 1.0 (exp (- z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+77) {
		tmp = exp((x - z));
	} else if (z <= 1e+19) {
		tmp = exp(fma(y, log(y), x));
	} else {
		tmp = 1.0 / exp((z - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+77)
		tmp = exp(Float64(x - z));
	elseif (z <= 1e+19)
		tmp = exp(fma(y, log(y), x));
	else
		tmp = Float64(1.0 / exp(Float64(z - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+77], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1e+19], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Exp[N[(z - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9000000000000001e77

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 100.0

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x - z}} \]

   if -1.9000000000000001e77 < z < 1e19

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x + y \cdot \log y}} \]

      2. +-commutative N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y + x}} \]

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 99.2

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

   5. Simplified 99.2%

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

   if 1e19 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 88.9

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{e^{x - z}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{x}}{e^{z}}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{z}}{e^{x}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{z}}{e^{x}}}} \]

      4. div-exp N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{{z}^{\color{blue}{\left(3 - 2\right)}} - x}} \]

      7. pow-div N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{{z}^{3}}{{z}^{2}}} - x}} \]

      8. sqr-pow N/A

         \[\leadsto \frac{1}{e^{\frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{{z}^{2}} - x}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{1}{e^{\frac{\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}}{{z}^{2}} - x}} \]

      10. sqr-neg N/A

          \[\leadsto \frac{1}{e^{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{{z}^{2}} - x}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{{z}^{2}} - x}} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{{z}^{2}} - x}} \]

      13. unpow-prod-down N/A

          \[\leadsto \frac{1}{e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}}{{z}^{2}} - x}} \]

      14. sqr-pow N/A

          \[\leadsto \frac{1}{e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}}{{z}^{2}} - x}} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{e^{\frac{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}{\color{blue}{z \cdot z}} - x}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}}{z \cdot z} - x}} \]

      17. cube-neg N/A

          \[\leadsto \frac{1}{e^{\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{z \cdot z} - x}} \]

      18. neg-sub0 N/A

          \[\leadsto \frac{1}{e^{\frac{\color{blue}{0 - {z}^{3}}}{z \cdot z} - x}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{\color{blue}{{0}^{3}} - {z}^{3}}{z \cdot z} - x}} \]

      20. +-lft-identity N/A

          \[\leadsto \frac{1}{e^{\frac{{0}^{3} - {z}^{3}}{z \cdot \color{blue}{\left(0 + z\right)}} - x}} \]

      21. +-commutative N/A

          \[\leadsto \frac{1}{e^{\frac{{0}^{3} - {z}^{3}}{z \cdot \color{blue}{\left(z + 0\right)}} - x}} \]

      22. distribute-rgt-out N/A

          \[\leadsto \frac{1}{e^{\frac{{0}^{3} - {z}^{3}}{\color{blue}{z \cdot z + 0 \cdot z}} - x}} \]

      23. +-lft-identity N/A

          \[\leadsto \frac{1}{e^{\frac{{0}^{3} - {z}^{3}}{\color{blue}{0 + \left(z \cdot z + 0 \cdot z\right)}} - x}} \]

      24. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{{0}^{3} - {z}^{3}}{\color{blue}{0 \cdot 0} + \left(z \cdot z + 0 \cdot z\right)} - x}} \]

      25. flip3-- N/A

          \[\leadsto \frac{1}{e^{\color{blue}{\left(0 - z\right)} - x}} \]

      26. neg-sub0 N/A

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

      27. lift-neg.f64 N/A

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

   7. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{1}{e^{z - x}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{z \cdot \left(z \cdot z\right)}}{z}, \mathsf{fma}\left(z, z \cdot 0.25, \mathsf{fma}\left(z, -0.5, 1\right)\right), \frac{-1}{\mathsf{fma}\left(z, 0.5, 1\right)}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 2e+34)
   (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
   (fma
    z
    (fma
     (/ (+ 2.0 (/ -16.0 (* z (* z z)))) z)
     (fma z (* z 0.25) (fma z -0.5 1.0))
     (/ -1.0 (fma z 0.5 1.0)))
    1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= 2e+34) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else {
		tmp = fma(z, fma(((2.0 + (-16.0 / (z * (z * z)))) / z), fma(z, (z * 0.25), fma(z, -0.5, 1.0)), (-1.0 / fma(z, 0.5, 1.0))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * log(y))) - z) <= 2e+34)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	else
		tmp = fma(z, fma(Float64(Float64(2.0 + Float64(-16.0 / Float64(z * Float64(z * z)))) / z), fma(z, Float64(z * 0.25), fma(z, -0.5, 1.0)), Float64(-1.0 / fma(z, 0.5, 1.0))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 2e+34], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(N[(2.0 + N[(-16.0 / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(z * N[(z * 0.25), $MachinePrecision] + N[(z * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.99999999999999989e34

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 62.2

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 62.2%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 25.3

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 25.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 25.3

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 25.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 61.0

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 61.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if 1.99999999999999989e34 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 37.1

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 37.1%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 33.5

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}{\color{blue}{\frac{{\left(z \cdot \frac{1}{2}\right)}^{3} - {-1}^{3}}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right) + \left(-1 \cdot -1 + \left(z \cdot \frac{1}{2}\right) \cdot -1\right)}}} + \left(\mathsf{neg}\left(\frac{1}{z \cdot \frac{1}{2} - -1}\right)\right), 1\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}{{\left(z \cdot \frac{1}{2}\right)}^{3} - {-1}^{3}} \cdot \left(\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right) + \left(-1 \cdot -1 + \left(z \cdot \frac{1}{2}\right) \cdot -1\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{z \cdot \frac{1}{2} - -1}\right)\right), 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\frac{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}{{\left(z \cdot \frac{1}{2}\right)}^{3} - {-1}^{3}}, \left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right) + \left(-1 \cdot -1 + \left(z \cdot \frac{1}{2}\right) \cdot -1\right), \mathsf{neg}\left(\frac{1}{z \cdot \frac{1}{2} - -1}\right)\right)}, 1\right) \]

   10. Applied egg-rr 2.2%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\frac{\left(z \cdot z\right) \cdot 0.25}{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), 0.125, 1\right)}, \mathsf{fma}\left(z, z \cdot 0.25, \mathsf{fma}\left(z, -0.5, 1\right)\right), -\frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}\right)}, 1\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{\frac{2 - 16 \cdot \frac{1}{{z}^{3}}}{z}}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{\frac{2 - 16 \cdot \frac{1}{{z}^{3}}}{z}}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{\color{blue}{2 + \left(\mathsf{neg}\left(16 \cdot \frac{1}{{z}^{3}}\right)\right)}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       3. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{\color{blue}{2 + \left(\mathsf{neg}\left(16 \cdot \frac{1}{{z}^{3}}\right)\right)}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \left(\mathsf{neg}\left(\color{blue}{\frac{16 \cdot 1}{{z}^{3}}}\right)\right)}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \left(\mathsf{neg}\left(\frac{\color{blue}{16}}{{z}^{3}}\right)\right)}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \color{blue}{\frac{\mathsf{neg}\left(16\right)}{{z}^{3}}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \color{blue}{\frac{\mathsf{neg}\left(16\right)}{{z}^{3}}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{\color{blue}{-16}}{{z}^{3}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{\color{blue}{z \cdot \left(z \cdot z\right)}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{z \cdot \color{blue}{{z}^{2}}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{\color{blue}{z \cdot {z}^{2}}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{z \cdot \color{blue}{\left(z \cdot z\right)}}}{z}, \mathsf{fma}\left(z, z \cdot \frac{1}{4}, \mathsf{fma}\left(z, \frac{-1}{2}, 1\right)\right), \mathsf{neg}\left(\frac{1}{\mathsf{fma}\left(z, \frac{1}{2}, 1\right)}\right)\right), 1\right) \]

       13. lower-*.f64 50.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{z \cdot \color{blue}{\left(z \cdot z\right)}}}{z}, \mathsf{fma}\left(z, z \cdot 0.25, \mathsf{fma}\left(z, -0.5, 1\right)\right), -\frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}\right), 1\right) \]

   13. Simplified 50.9%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{\frac{2 + \frac{-16}{z \cdot \left(z \cdot z\right)}}{z}}, \mathsf{fma}\left(z, z \cdot 0.25, \mathsf{fma}\left(z, -0.5, 1\right)\right), -\frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(\frac{2 + \frac{-16}{z \cdot \left(z \cdot z\right)}}{z}, \mathsf{fma}\left(z, z \cdot 0.25, \mathsf{fma}\left(z, -0.5, 1\right)\right), \frac{-1}{\mathsf{fma}\left(z, 0.5, 1\right)}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -2 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) -2e+34)
   (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))
   (fma (* z (fma (* z z) 0.25 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= -2e+34) {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	} else {
		tmp = fma((z * fma((z * z), 0.25, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * log(y))) - z) <= -2e+34)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	else
		tmp = fma(Float64(z * fma(Float64(z * z), 0.25, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], -2e+34], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(z * z), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1.99999999999999989e34

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 60.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 60.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 2.2

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 2.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 2.2

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 2.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 58.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 58.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]

   if -1.99999999999999989e34 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 43.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 43.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 40.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 40.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right) \cdot z} + 1 \]

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied egg-rr 44.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot z, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -200:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x (* y (log y))) -200.0) (* z (* z 0.5)) (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + (y * log(y))) <= -200.0) {
		tmp = z * (z * 0.5);
	} else {
		tmp = fma(z, (z * 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(y * log(y))) <= -200.0)
		tmp = Float64(z * Float64(z * 0.5));
	else
		tmp = fma(z, Float64(z * 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -200.0], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -200

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 34.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 34.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 12.0

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 12.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 36.5

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -200 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 52.1

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 52.1%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 34.2

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 34.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}}, 1\right) \]

       2. lower-*.f64 34.2

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]

   11. Simplified 34.2%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+107}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+154)
   (* z (* z 0.5))
   (if (<= z -1.15e+77)
     (/ (fma 0.25 (* (* z z) (* z z)) -1.0) (fma z (* z 0.5) -1.0))
     (if (<= z 6.5e+107)
       (exp x)
       (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+154) {
		tmp = z * (z * 0.5);
	} else if (z <= -1.15e+77) {
		tmp = fma(0.25, ((z * z) * (z * z)), -1.0) / fma(z, (z * 0.5), -1.0);
	} else if (z <= 6.5e+107) {
		tmp = exp(x);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+154)
		tmp = Float64(z * Float64(z * 0.5));
	elseif (z <= -1.15e+77)
		tmp = Float64(fma(0.25, Float64(Float64(z * z) * Float64(z * z)), -1.0) / fma(z, Float64(z * 0.5), -1.0));
	elseif (z <= 6.5e+107)
		tmp = exp(x);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+154], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.15e+77], N[(N[(0.25 * N[(N[(z * z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(z * N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+107], N[Exp[x], $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.8999999999999999e154

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 89.6

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 89.6%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 89.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 89.6

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 89.6%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -1.8999999999999999e154 < z < -1.14999999999999997e77

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 90.2

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 90.2%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 5.7

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 5.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}}, 1\right) \]

       2. lower-*.f64 5.7

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]

   11. Simplified 5.7%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. sub-neg N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. unpow-prod-down N/A

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

       12. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{\frac{1}{4}} \cdot {\left(z \cdot z\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       13. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{4} \cdot {\left(z \cdot z\right)}^{2} + \color{blue}{-1}}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       14. lower-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, {\left(z \cdot z\right)}^{2}, -1\right)}}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       15. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(z \cdot z\right) \cdot \left(z \cdot z\right)}, -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\left(z \cdot z\right) \cdot \left(z \cdot z\right)}, -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) - 1} \]

       17. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\color{blue}{z \cdot \left(z \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

       18. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{4}, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{z \cdot \left(z \cdot \frac{1}{2}\right) + \color{blue}{-1}} \]

       19. lower-fma.f64 90.2

           \[\leadsto \frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\color{blue}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}} \]

   13. Applied egg-rr 90.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.25, \left(z \cdot z\right) \cdot \left(z \cdot z\right), -1\right)}{\mathsf{fma}\left(z, z \cdot 0.5, -1\right)}} \]

   if -1.14999999999999997e77 < z < 6.5000000000000006e107

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 62.0

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 57.1

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

   8. Simplified 57.1%

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

   if 6.5000000000000006e107 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 76.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 76.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 21.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 21.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 21.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 76.5

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 76.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 90.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+66}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e+66) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+66) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, 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 (y <= 1.7d+66) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+66) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.7e+66:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e+66)
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.7e+66)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e+66], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000015e66

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 91.9

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 91.9%

      \[\leadsto \color{blue}{e^{x - z}} \]

   if 1.70000000000000015e66 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \]

      6. lower-log.f64 85.8

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

   5. Simplified 85.8%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{y \cdot \color{blue}{\log y}} \]

      2. *-commutative N/A

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\color{blue}{\log y} \cdot y} \]

      4. pow-to-exp N/A

         \[\leadsto \color{blue}{{y}^{y}} \]

      5. lift-pow.f64 85.8

         \[\leadsto \color{blue}{{y}^{y}} \]

   7. Applied egg-rr 85.8%

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

Alternative 15: 51.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+88)
   (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= z 3.15e+106)
     (fma x (fma x 0.5 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z (* z -0.25) 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+88) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (z <= 3.15e+106) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, (z * -0.25), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+88)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (z <= 3.15e+106)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, Float64(z * -0.25), 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+88], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 3.15e+106], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * N[(z * -0.25), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1000000000000001e88

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 89.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 89.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. lower-fma.f64 81.7

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if -3.1000000000000001e88 < z < 3.14999999999999987e106

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 61.6

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 61.6%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 56.8

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

   8. Simplified 56.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 32.4

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if 3.14999999999999987e106 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 76.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 76.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 21.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 21.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 21.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 21.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right)\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}\right) + 1}, 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{4} \cdot {z}^{2}, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{4} \cdot {z}^{2} + \frac{1}{2}}, 1\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{-1}{4} \cdot \color{blue}{\left(z \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\left(\frac{-1}{4} \cdot z\right) \cdot z} + \frac{1}{2}, 1\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{-1}{4} \cdot z\right)} + \frac{1}{2}, 1\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{-1}{4} \cdot z, \frac{1}{2}\right)}, 1\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{4}}, \frac{1}{2}\right), 1\right), 1\right)} \]

       11. lower-*.f64 76.5

           \[\leadsto \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot -0.25}, 0.5\right), 1\right), 1\right)} \]

   13. Simplified 76.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot -0.25, 0.5\right), 1\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+88)
   (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= z 5e+154)
     (fma x (fma x 0.5 1.0) 1.0)
     (/ 1.0 (fma z (fma 0.5 z 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+88) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (z <= 5e+154) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(0.5, z, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+88)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (z <= 5e+154)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(0.5, z, 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+88], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 5e+154], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(0.5 * z + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1000000000000001e88

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 89.5

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 89.5%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. lower-fma.f64 81.7

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if -3.1000000000000001e88 < z < 5.00000000000000004e154

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 62.5

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 62.5%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 55.1

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

   8. Simplified 55.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 31.4

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   11. Simplified 31.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if 5.00000000000000004e154 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 78.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 78.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 22.8

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 22.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2}, -1\right)} + 1 \]

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \mathsf{fma}\left(z, \frac{1}{2}, -1\right) + 1}}} \]

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 22.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   10. Applied egg-rr 22.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}}} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, 1 + \frac{1}{2} \cdot z, 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + 1}, 1\right)} \]

       4. lower-fma.f64 78.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, 1\right)}, 1\right)} \]

   13. Simplified 78.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -185:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -185.0)
   (* z (* z 0.5))
   (if (<= x 9e+102)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -185.0) {
		tmp = z * (z * 0.5);
	} else if (x <= 9e+102) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -185.0)
		tmp = Float64(z * Float64(z * 0.5));
	elseif (x <= 9e+102)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -185.0], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e+102], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -185

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 33.8

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 33.8%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 14.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 32.4

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -185 < x < 9.00000000000000042e102

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 59.9

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 59.9%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{-1}, 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{-1}{6} \cdot z, -1\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot z + \frac{1}{2}}, -1\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

      8. lower-fma.f64 37.0

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 37.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if 9.00000000000000042e102 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 97.4

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 97.4%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 87.0

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

   8. Simplified 87.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

       7. lower-fma.f64 87.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   11. Simplified 87.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -210.0)
   (* z (* z 0.5))
   (if (<= x 9.5e+102)
     (fma z (fma z 0.5 -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -210.0) {
		tmp = z * (z * 0.5);
	} else if (x <= 9.5e+102) {
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -210.0)
		tmp = Float64(z * Float64(z * 0.5));
	elseif (x <= 9.5e+102)
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -210.0], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+102], N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -210

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 33.8

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 33.8%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 14.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 32.4

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -210 < x < 9.4999999999999992e102

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 59.9

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 59.9%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 36.6

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 36.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   if 9.4999999999999992e102 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 97.4

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 97.4%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 87.0

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

   8. Simplified 87.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{1}{6} \cdot x, 1\right)}, 1\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, 1\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

       7. lower-fma.f64 87.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   11. Simplified 87.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 42.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -210.0)
   (* z (* z 0.5))
   (if (<= x 1.95e+142)
     (fma z (fma z 0.5 -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -210.0) {
		tmp = z * (z * 0.5);
	} else if (x <= 1.95e+142) {
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -210.0)
		tmp = Float64(z * Float64(z * 0.5));
	elseif (x <= 1.95e+142)
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -210.0], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+142], N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -210

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 33.8

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 33.8%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 14.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 32.4

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -210 < x < 1.95e142

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 59.0

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 59.0%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 35.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   if 1.95e142 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 100.0

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 90.8

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

   8. Simplified 90.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 85.1

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   11. Simplified 85.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 41.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -210.0)
   (* z (* z 0.5))
   (if (<= x 1.95e+142) (fma z (* z 0.5) 1.0) (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -210.0) {
		tmp = z * (z * 0.5);
	} else if (x <= 1.95e+142) {
		tmp = fma(z, (z * 0.5), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -210.0)
		tmp = Float64(z * Float64(z * 0.5));
	elseif (x <= 1.95e+142)
		tmp = fma(z, Float64(z * 0.5), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -210.0], N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+142], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -210

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 33.8

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 33.8%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 14.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot z} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z\right)} \]

       5. *-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{1}{2}\right)} \]

       6. lower-*.f64 32.4

          \[\leadsto z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot 0.5\right)} \]

   if -210 < x < 1.95e142

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 59.0

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 59.0%

      \[\leadsto e^{\color{blue}{-z}} \]

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, z \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 35.9

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5, -1\right)}, 1\right) \]

   8. Simplified 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{2}}, 1\right) \]

       2. lower-*.f64 35.8

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]

   11. Simplified 35.8%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 0.5}, 1\right) \]

   if 1.95e142 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{x - z}} \]

      2. lower--.f64 100.0

         \[\leadsto e^{\color{blue}{x - z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x - z}} \]

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 90.8

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

   8. Simplified 90.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 85.1

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   11. Simplified 85.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 15.0% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 1 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 1.0 z))

C

double code(double x, double y, double z) {
	return 1.0 - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 - z
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 - z;
}

Python

def code(x, y, z):
	return 1.0 - z

Julia

function code(x, y, z)
	return Float64(1.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(1.0 - z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

   2. lower-neg.f64 48.6

      \[\leadsto e^{\color{blue}{-z}} \]

5. Simplified 48.6%

   \[\leadsto e^{\color{blue}{-z}} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{1 + -1 \cdot z} \]
7. Step-by-step derivation

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{1 - z} \]

   3. lower--.f64 13.7

      \[\leadsto \color{blue}{1 - z} \]

8. Simplified 13.7%

   \[\leadsto \color{blue}{1 - z} \]
9. Add Preprocessing

Alternative 22: 14.7% accurate, 212.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

   2. lower-neg.f64 48.6

      \[\leadsto e^{\color{blue}{-z}} \]

5. Simplified 48.6%

   \[\leadsto e^{\color{blue}{-z}} \]

6. Taylor expanded in z around 0

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

8. Simplified 13.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x - z\right) + \log y \cdot y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return exp(((x - z) + (log(y) * 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 = exp(((x - z) + (log(y) * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024208 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (exp (+ (- x z) (* (log y) y))))

  (exp (- (+ x (* y (log y))) z)))