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

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 23

Speedup: 1.0×

• 57.8% 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: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(-0.25, z \cdot z, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -2e-10)
     (exp x)
     (if (<= t_0 -1e-122)
       (/ 1.0 (fma z (fma z (fma -0.25 (* z z) 0.5) 1.0) 1.0))
       (if (<= t_0 5e-39)
         (fma
          (fma z 0.5 -1.0)
          (* z (* (fma z 0.5 -1.0) (* z (* z (fma z 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 <= -2e-10) {
		tmp = exp(x);
	} else if (t_0 <= -1e-122) {
		tmp = 1.0 / fma(z, fma(z, fma(-0.25, (z * z), 0.5), 1.0), 1.0);
	} else if (t_0 <= 5e-39) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (fma(z, 0.5, -1.0) * (z * (z * fma(z, 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 <= -2e-10)
		tmp = exp(x);
	elseif (t_0 <= -1e-122)
		tmp = Float64(1.0 / fma(z, fma(z, fma(-0.25, Float64(z * z), 0.5), 1.0), 1.0));
	elseif (t_0 <= 5e-39)
		tmp = fma(fma(z, 0.5, -1.0), Float64(z * Float64(fma(z, 0.5, -1.0) * Float64(z * Float64(z * fma(z, 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, -2e-10], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, -1e-122], N[(1.0 / N[(z * N[(z * N[(-0.25 * N[(z * z), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-39], N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -2.00000000000000007e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 87.8

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

   8. Simplified 87.8%

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

   if -2.00000000000000007e-10 < (+.f64 x (*.f64 y (log.f64 y))) < -1.00000000000000006e-122

   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 99.0

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

   5. Simplified 99.0%

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

   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 54.3

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

   8. Simplified 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 54.3

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

   10. Applied egg-rr 54.3%

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

   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. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 72.4

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

   13. Simplified 72.4%

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

   if -1.00000000000000006e-122 < (+.f64 x (*.f64 y (log.f64 y))) < 4.9999999999999998e-39

   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

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   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 80.3

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

   8. Simplified 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 58.9%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 86.8%

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

   if 4.9999999999999998e-39 < (+.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 74.5

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

   5. Simplified 74.5%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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. exp-to-pow N/A

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

      5. lower-pow.f64 74.5

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

   7. Applied egg-rr 74.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -2 \cdot 10^{-10}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x + y \cdot \log y \leq -1 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(-0.25, z \cdot z, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;x + y \cdot \log y \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+23}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+29}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -4e+23) (exp x) (if (<= t_0 1e+29) (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 <= -4e+23) {
		tmp = exp(x);
	} else if (t_0 <= 1e+29) {
		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 <= (-4d+23)) then
        tmp = exp(x)
    else if (t_0 <= 1d+29) 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 <= -4e+23) {
		tmp = Math.exp(x);
	} else if (t_0 <= 1e+29) {
		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 <= -4e+23:
		tmp = math.exp(x)
	elif t_0 <= 1e+29:
		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 <= -4e+23)
		tmp = exp(x);
	elseif (t_0 <= 1e+29)
		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 <= -4e+23)
		tmp = exp(x);
	elseif (t_0 <= 1e+29)
		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, -4e+23], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e+29], 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))) < -3.9999999999999997e23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 93.3

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

   8. Simplified 93.3%

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

   if -3.9999999999999997e23 < (+.f64 x (*.f64 y (log.f64 y))) < 9.99999999999999914e28

   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 92.2

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

   5. Simplified 92.2%

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

   if 9.99999999999999914e28 < (+.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 76.6

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

   5. Simplified 76.6%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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. exp-to-pow N/A

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

      5. lower-pow.f64 76.6

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

   7. Applied egg-rr 76.6%

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

Alternative 4: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 20:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 20.0) (exp (- x z)) (exp (fma y (log y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 20.0) {
		tmp = 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)) <= 20.0)
		tmp = 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], 20.0], N[Exp[N[(x - z), $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)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 98.7

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

   5. Simplified 98.7%

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

   if 20 < (*.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 91.4

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

   5. Simplified 91.4%

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

Alternative 5: 31.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \left(z \cdot z\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+85}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* (* z z) 0.5)))
   (if (<= t_0 -2e+127) t_1 (if (<= t_0 2e+85) (+ x 1.0) 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 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 2e+85) {
		tmp = x + 1.0;
	} 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 <= (-2d+127)) then
        tmp = t_1
    else if (t_0 <= 2d+85) then
        tmp = x + 1.0d0
    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 <= -2e+127) {
		tmp = t_1;
	} else if (t_0 <= 2e+85) {
		tmp = x + 1.0;
	} 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 <= -2e+127:
		tmp = t_1
	elif t_0 <= 2e+85:
		tmp = x + 1.0
	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(Float64(z * z) * 0.5)
	tmp = 0.0
	if (t_0 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 2e+85)
		tmp = Float64(x + 1.0);
	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 <= -2e+127)
		tmp = t_1;
	elseif (t_0 <= 2e+85)
		tmp = x + 1.0;
	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[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+127], t$95$1, If[LessEqual[t$95$0, 2e+85], N[(x + 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1.99999999999999991e127 or 2e85 < (-.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 44.8

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

   5. Simplified 44.8%

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

   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 20.1

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

   8. Simplified 20.1%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 25.2

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

   11. Simplified 25.2%

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

   if -1.99999999999999991e127 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2e85

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 86.5

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

   5. Simplified 86.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 66.6

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

   8. Simplified 66.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 48.8

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

   11. Simplified 48.8%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+85}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 5 \cdot 10^{+127}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 5e+127) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 5e+127) {
		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 * log(y)) <= 5d+127) 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 * Math.log(y)) <= 5e+127) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * math.log(y)) <= 5e+127:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 5e+127)
		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 * log(y)) <= 5e+127)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 5e+127], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 5.0000000000000004e127

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 92.2

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

   5. Simplified 92.2%

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

   if 5.0000000000000004e127 < (*.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 93.3

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

   5. Simplified 93.3%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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. exp-to-pow N/A

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

      5. lower-pow.f64 93.3

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

   7. Applied egg-rr 93.3%

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

Alternative 7: 53.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 0.002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(-0.25, z \cdot z, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 0.002)
   (/ 1.0 (fma z (fma z (fma -0.25 (* z z) 0.5) 1.0) 1.0))
   (fma
    (fma z 0.5 -1.0)
    (* z (* (fma z 0.5 -1.0) (* z (* z (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) <= 0.002) {
		tmp = 1.0 / fma(z, fma(z, fma(-0.25, (z * z), 0.5), 1.0), 1.0);
	} else {
		tmp = fma(fma(z, 0.5, -1.0), (z * (fma(z, 0.5, -1.0) * (z * (z * 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) <= 0.002)
		tmp = Float64(1.0 / fma(z, fma(z, fma(-0.25, Float64(z * z), 0.5), 1.0), 1.0));
	else
		tmp = fma(fma(z, 0.5, -1.0), Float64(z * Float64(fma(z, 0.5, -1.0) * Float64(z * Float64(z * 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], 0.002], N[(1.0 / N[(z * N[(z * N[(-0.25 * N[(z * z), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2e-3

   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 71.7

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

   5. Simplified 71.7%

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

   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 29.4

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

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 29.4

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

   10. Applied egg-rr 29.4%

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

   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. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 71.5

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

   13. Simplified 71.5%

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

   if 2e-3 < (-.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 37.0

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

   5. Simplified 37.0%

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

   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 25.0

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

   8. Simplified 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 6.9%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 48.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 0.002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(-0.25, z \cdot z, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;z \leq 2150000000000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;e^{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.6e+55)
   (fma
    (fma z 0.5 -1.0)
    (* z (* (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0)))))
    1.0)
   (if (<= z 2150000000000.0)
     (exp x)
     (if (<= z 6.8e+155) (exp z) (/ 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 <= -3.6e+55) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (fma(z, 0.5, -1.0) * (z * (z * fma(z, 0.5, -1.0))))), 1.0);
	} else if (z <= 2150000000000.0) {
		tmp = exp(x);
	} else if (z <= 6.8e+155) {
		tmp = exp(z);
	} 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 <= -3.6e+55)
		tmp = fma(fma(z, 0.5, -1.0), Float64(z * Float64(fma(z, 0.5, -1.0) * Float64(z * Float64(z * fma(z, 0.5, -1.0))))), 1.0);
	elseif (z <= 2150000000000.0)
		tmp = exp(x);
	elseif (z <= 6.8e+155)
		tmp = exp(z);
	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, -3.6e+55], N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2150000000000.0], N[Exp[x], $MachinePrecision], If[LessEqual[z, 6.8e+155], N[Exp[z], $MachinePrecision], N[(1.0 / N[(z * N[(0.5 * z + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.59999999999999987e55

   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 92.1

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

   5. Simplified 92.1%

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

   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 56.9

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

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 4.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 92.1%

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

   if -3.59999999999999987e55 < z < 2.15e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 64.4

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

   8. Simplified 64.4%

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

   if 2.15e12 < z < 6.8000000000000002e155

   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 34.4

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

   5. Simplified 34.4%

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

      1. neg-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. neg-sub0 N/A

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

      5. cube-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. sqr-pow N/A

         \[\leadsto e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      8. unpow-prod-down N/A

         \[\leadsto e^{\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto e^{\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      10. lift-neg.f64 N/A

          \[\leadsto e^{\frac{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      11. sqr-neg N/A

          \[\leadsto e^{\frac{{\color{blue}{\left(z \cdot z\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      12. pow-prod-down N/A

          \[\leadsto e^{\frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      13. sqr-pow N/A

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

      14. metadata-eval N/A

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

      15. +-lft-identity N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. +-lft-identity N/A

          \[\leadsto e^{\frac{{z}^{3}}{z \cdot \color{blue}{z}}} \]

      19. pow2 N/A

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

      20. pow-div N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lower-exp.f64 67.2

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

   7. Applied egg-rr 67.2%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;z \leq 2150000000000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;e^{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (fma z 0.5 -1.0)
          (* z (* (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0)))))
          1.0)))
   (if (<= z -3.6e+55)
     t_0
     (if (<= z 3.3e+51)
       (exp x)
       (if (<= z 6.8e+155) t_0 (/ 1.0 (fma z (fma 0.5 z 1.0) 1.0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(fma(z, 0.5, -1.0), (z * (fma(z, 0.5, -1.0) * (z * (z * fma(z, 0.5, -1.0))))), 1.0);
	double tmp;
	if (z <= -3.6e+55) {
		tmp = t_0;
	} else if (z <= 3.3e+51) {
		tmp = exp(x);
	} else if (z <= 6.8e+155) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(z, fma(0.5, z, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(fma(z, 0.5, -1.0), Float64(z * Float64(fma(z, 0.5, -1.0) * Float64(z * Float64(z * fma(z, 0.5, -1.0))))), 1.0)
	tmp = 0.0
	if (z <= -3.6e+55)
		tmp = t_0;
	elseif (z <= 3.3e+51)
		tmp = exp(x);
	elseif (z <= 6.8e+155)
		tmp = t_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_] := Block[{t$95$0 = N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -3.6e+55], t$95$0, If[LessEqual[z, 3.3e+51], N[Exp[x], $MachinePrecision], If[LessEqual[z, 6.8e+155], t$95$0, 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 < -3.59999999999999987e55 or 3.2999999999999997e51 < z < 6.8000000000000002e155

   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 72.2

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

   5. Simplified 72.2%

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

   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 39.6

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

   8. Simplified 39.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 7.7%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 82.3%

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

   if -3.59999999999999987e55 < z < 3.2999999999999997e51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 68.8

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

   5. Simplified 68.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 63.6

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

   8. Simplified 63.6%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-275}:\\ \;\;\;\;\frac{t\_0}{\left(z \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0.25}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (fma z 0.5 -1.0)
          (* z (* (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0)))))
          1.0)))
   (if (<= z -5e-83)
     t_0
     (if (<= z 2.75e-275)
       (/ t_0 (* (* z (* z (* z z))) 0.25))
       (if (<= z 9e+45)
         (fma (* x (fma (* x x) 0.25 -1.0)) (/ 1.0 (fma x 0.5 -1.0)) 1.0)
         (if (<= z 6.8e+155) t_0 (/ 1.0 (fma z (fma 0.5 z 1.0) 1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(fma(z, 0.5, -1.0), (z * (fma(z, 0.5, -1.0) * (z * (z * fma(z, 0.5, -1.0))))), 1.0);
	double tmp;
	if (z <= -5e-83) {
		tmp = t_0;
	} else if (z <= 2.75e-275) {
		tmp = t_0 / ((z * (z * (z * z))) * 0.25);
	} else if (z <= 9e+45) {
		tmp = fma((x * fma((x * x), 0.25, -1.0)), (1.0 / fma(x, 0.5, -1.0)), 1.0);
	} else if (z <= 6.8e+155) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(z, fma(0.5, z, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(fma(z, 0.5, -1.0), Float64(z * Float64(fma(z, 0.5, -1.0) * Float64(z * Float64(z * fma(z, 0.5, -1.0))))), 1.0)
	tmp = 0.0
	if (z <= -5e-83)
		tmp = t_0;
	elseif (z <= 2.75e-275)
		tmp = Float64(t_0 / Float64(Float64(z * Float64(z * Float64(z * z))) * 0.25));
	elseif (z <= 9e+45)
		tmp = fma(Float64(x * fma(Float64(x * x), 0.25, -1.0)), Float64(1.0 / fma(x, 0.5, -1.0)), 1.0);
	elseif (z <= 6.8e+155)
		tmp = t_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_] := Block[{t$95$0 = N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -5e-83], t$95$0, If[LessEqual[z, 2.75e-275], N[(t$95$0 / N[(N[(z * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e+45], N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.8e+155], t$95$0, N[(1.0 / N[(z * N[(0.5 * z + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5e-83 or 8.9999999999999997e45 < z < 6.8000000000000002e155

   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 68.8

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

   5. Simplified 68.8%

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

   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 35.4

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

   8. Simplified 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 12.4%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 70.4%

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

   if -5e-83 < z < 2.74999999999999994e-275

   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 15.9

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

   5. Simplified 15.9%

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

   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 15.9

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

   8. Simplified 15.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

   10. Applied egg-rr 15.9%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 64.1

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

   13. Simplified 64.1%

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

   if 2.74999999999999994e-275 < z < 8.9999999999999997e45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 64.6

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

   5. Simplified 64.6%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 63.3

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

   8. Simplified 63.3%

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

   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. lower-fma.f64 45.6

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

   11. Simplified 45.6%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. div-inv N/A

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

       7. lower-fma.f64 N/A

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

   13. Applied egg-rr 46.9%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-275}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)}{\left(z \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0.25}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(\mathsf{fma}\left(z, 0.5, -1\right) \cdot \left(z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, z \cdot 0.25, -1\right), \frac{-1}{\mathsf{fma}\left(z, -0.5, -1\right)}, 1\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+100)
   (fma (* z (fma z (* z 0.25) -1.0)) (/ -1.0 (fma z -0.5 -1.0)) 1.0)
   (if (<= z -1.2e+56)
     (/
      (fma (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0))) -1.0)
      (fma z (fma z 0.5 -1.0) -1.0))
     (if (<= z 6.8e+155)
       (fma (* x (fma (* x x) 0.25 -1.0)) (/ 1.0 (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 <= -2e+100) {
		tmp = fma((z * fma(z, (z * 0.25), -1.0)), (-1.0 / fma(z, -0.5, -1.0)), 1.0);
	} else if (z <= -1.2e+56) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (z * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else if (z <= 6.8e+155) {
		tmp = fma((x * fma((x * x), 0.25, -1.0)), (1.0 / 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 <= -2e+100)
		tmp = fma(Float64(z * fma(z, Float64(z * 0.25), -1.0)), Float64(-1.0 / fma(z, -0.5, -1.0)), 1.0);
	elseif (z <= -1.2e+56)
		tmp = Float64(fma(fma(z, 0.5, -1.0), Float64(z * Float64(z * fma(z, 0.5, -1.0))), -1.0) / fma(z, fma(z, 0.5, -1.0), -1.0));
	elseif (z <= 6.8e+155)
		tmp = fma(Float64(x * fma(Float64(x * x), 0.25, -1.0)), Float64(1.0 / 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, -2e+100], N[(N[(z * N[(z * N[(z * 0.25), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(z * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, -1.2e+56], N[(N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * 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], If[LessEqual[z, 6.8e+155], N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * 0.5 + -1.0), $MachinePrecision]), $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 4 regimes

2. if z < -2.00000000000000003e100

   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 90.4

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

   5. Simplified 90.4%

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

   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 68.3

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

   8. Simplified 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. *-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 90.4%

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

   if -2.00000000000000003e100 < z < -1.20000000000000007e56

   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

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   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 5.0

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

   8. Simplified 5.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 78.7

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

   10. Applied egg-rr 78.7%

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

   if -1.20000000000000007e56 < z < 6.8000000000000002e155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 60.4

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

   8. Simplified 60.4%

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

   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. lower-fma.f64 37.3

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

   11. Simplified 37.3%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. div-inv N/A

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

       7. lower-fma.f64 N/A

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

   13. Applied egg-rr 41.0%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, z \cdot 0.25, -1\right), \frac{-1}{\mathsf{fma}\left(z, -0.5, -1\right)}, 1\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+56)
   (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= z 6.8e+155)
     (fma (* x (fma (* x x) 0.25 -1.0)) (/ 1.0 (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 <= -1.2e+56) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (z <= 6.8e+155) {
		tmp = fma((x * fma((x * x), 0.25, -1.0)), (1.0 / 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 <= -1.2e+56)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (z <= 6.8e+155)
		tmp = fma(Float64(x * fma(Float64(x * x), 0.25, -1.0)), Float64(1.0 / 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, -1.2e+56], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.8e+155], N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * 0.5 + -1.0), $MachinePrecision]), $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 < -1.20000000000000007e56

   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 92.1

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

   5. Simplified 92.1%

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

   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 75.5

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

   8. Simplified 75.5%

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

   if -1.20000000000000007e56 < z < 6.8000000000000002e155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 60.4

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

   8. Simplified 60.4%

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

   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. lower-fma.f64 37.3

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

   11. Simplified 37.3%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. div-inv N/A

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

       7. lower-fma.f64 N/A

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

   13. Applied egg-rr 41.0%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(-0.25, z \cdot z, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+56)
   (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= z 2.55e+81)
     (fma x (fma x (fma 0.16666666666666666 x 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma -0.25 (* z z) 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+56) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (z <= 2.55e+81) {
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(-0.25, (z * z), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+56)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (z <= 2.55e+81)
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(-0.25, Float64(z * z), 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+56], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2.55e+81], N[(x * N[(x * N[(0.16666666666666666 * x + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(-0.25 * N[(z * z), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000007e56

   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 92.1

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

   5. Simplified 92.1%

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

   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 75.5

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

   8. Simplified 75.5%

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

   if -1.20000000000000007e56 < z < 2.5500000000000001e81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 63.1

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

   8. Simplified 63.1%

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

   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. lower-fma.f64 38.7

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

   11. Simplified 38.7%

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

   if 2.5500000000000001e81 < 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 69.5

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

   5. Simplified 69.5%

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

   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 13.2

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

   8. Simplified 13.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 13.2

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

   10. Applied egg-rr 13.2%

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

   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. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 69.5

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

   13. Simplified 69.5%

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

Alternative 14: 49.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+56)
   (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= z 6.8e+155)
     (fma x (fma x (fma 0.16666666666666666 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 <= -1.2e+56) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (z <= 6.8e+155) {
		tmp = fma(x, fma(x, fma(0.16666666666666666, 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 <= -1.2e+56)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (z <= 6.8e+155)
		tmp = fma(x, fma(x, fma(0.16666666666666666, 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, -1.2e+56], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.8e+155], N[(x * N[(x * N[(0.16666666666666666 * x + 0.5), $MachinePrecision] + 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 < -1.20000000000000007e56

   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 92.1

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

   5. Simplified 92.1%

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

   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 75.5

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

   8. Simplified 75.5%

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

   if -1.20000000000000007e56 < z < 6.8000000000000002e155

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 60.4

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

   8. Simplified 60.4%

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

   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. lower-fma.f64 37.4

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

   11. Simplified 37.4%

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

   if 6.8000000000000002e155 < 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 86.3

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

   5. Simplified 86.3%

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

   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 15.2

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

   8. Simplified 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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 15.2

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

   10. Applied egg-rr 15.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \frac{1}{\color{blue}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}} \]
   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 86.3

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

   13. Simplified 86.3%

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

Alternative 15: 44.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e+19)
   (* (* z z) 0.5)
   (if (<= x 8.5e+40)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
     (fma x (fma x (fma 0.16666666666666666 x 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+19) {
		tmp = (z * z) * 0.5;
	} else if (x <= 8.5e+40) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e+19)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 8.5e+40)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e+19], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8.5e+40], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.16666666666666666 * x + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3e19

   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 36.8

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

   5. Simplified 36.8%

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

   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 11.5

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

   8. Simplified 11.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   if -2.3e19 < x < 8.49999999999999996e40

   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 66.5

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

   5. Simplified 66.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   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 41.8

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

   8. Simplified 41.8%

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

   if 8.49999999999999996e40 < 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

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

      1. lower--.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 86.2

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

   8. Simplified 86.2%

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

   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. lower-fma.f64 68.1

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

   11. Simplified 68.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.15e+19)
   (* (* z z) 0.5)
   (if (<= x 4.5e+96)
     (- (fma z (* z 0.5) 1.0) z)
     (fma x (fma x (fma 0.16666666666666666 x 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = (z * z) * 0.5;
	} else if (x <= 4.5e+96) {
		tmp = fma(z, (z * 0.5), 1.0) - z;
	} else {
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.15e+19)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 4.5e+96)
		tmp = Float64(fma(z, Float64(z * 0.5), 1.0) - z);
	else
		tmp = fma(x, fma(x, fma(0.16666666666666666, x, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.15e+19], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.5e+96], N[(N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(x * N[(0.16666666666666666 * x + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15e19

   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 36.8

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

   5. Simplified 36.8%

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

   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 11.5

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

   8. Simplified 11.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   if -3.15e19 < x < 4.49999999999999957e96

   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 63.1

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

   5. Simplified 63.1%

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

   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 34.9

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

   8. Simplified 34.9%

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

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 34.9

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

   10. Applied egg-rr 34.9%

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

   if 4.49999999999999957e96 < 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

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

      1. lower--.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 85.0

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

   8. Simplified 85.0%

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

   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. lower-fma.f64 83.1

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

   11. Simplified 83.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot 0.5\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-139}:\\ \;\;\;\;1 - z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) 0.5)))
   (if (<= x -7.2e-147)
     t_0
     (if (<= x 1.1e-139)
       (- 1.0 z)
       (if (<= x 1.85e+152) t_0 (* x (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * z) * 0.5;
	double tmp;
	if (x <= -7.2e-147) {
		tmp = t_0;
	} else if (x <= 1.1e-139) {
		tmp = 1.0 - z;
	} else if (x <= 1.85e+152) {
		tmp = t_0;
	} else {
		tmp = x * (x * 0.5);
	}
	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 = (z * z) * 0.5d0
    if (x <= (-7.2d-147)) then
        tmp = t_0
    else if (x <= 1.1d-139) then
        tmp = 1.0d0 - z
    else if (x <= 1.85d+152) then
        tmp = t_0
    else
        tmp = x * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * z) * 0.5;
	double tmp;
	if (x <= -7.2e-147) {
		tmp = t_0;
	} else if (x <= 1.1e-139) {
		tmp = 1.0 - z;
	} else if (x <= 1.85e+152) {
		tmp = t_0;
	} else {
		tmp = x * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * z) * 0.5
	tmp = 0
	if x <= -7.2e-147:
		tmp = t_0
	elif x <= 1.1e-139:
		tmp = 1.0 - z
	elif x <= 1.85e+152:
		tmp = t_0
	else:
		tmp = x * (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * z) * 0.5)
	tmp = 0.0
	if (x <= -7.2e-147)
		tmp = t_0;
	elseif (x <= 1.1e-139)
		tmp = Float64(1.0 - z);
	elseif (x <= 1.85e+152)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * z) * 0.5;
	tmp = 0.0;
	if (x <= -7.2e-147)
		tmp = t_0;
	elseif (x <= 1.1e-139)
		tmp = 1.0 - z;
	elseif (x <= 1.85e+152)
		tmp = t_0;
	else
		tmp = x * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -7.2e-147], t$95$0, If[LessEqual[x, 1.1e-139], N[(1.0 - z), $MachinePrecision], If[LessEqual[x, 1.85e+152], t$95$0, N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.20000000000000023e-147 or 1.10000000000000005e-139 < x < 1.84999999999999998e152

   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 49.3

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

   5. Simplified 49.3%

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

   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 21.3

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

   8. Simplified 21.3%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 23.5

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

   11. Simplified 23.5%

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

   if -7.20000000000000023e-147 < x < 1.10000000000000005e-139

   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 69.2

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

   5. Simplified 69.2%

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

   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 33.2

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

   8. Simplified 33.2%

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

   if 1.84999999999999998e152 < 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

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

      1. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.9

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

   8. Simplified 89.9%

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

   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. lower-fma.f64 87.7

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

   11. Simplified 87.7%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 87.7

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

   14. Simplified 87.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-147}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-139}:\\ \;\;\;\;1 - z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.15e+19)
   (* (* z z) 0.5)
   (if (<= x 1.85e+152) (- (fma z (* z 0.5) 1.0) z) (* x (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = (z * z) * 0.5;
	} else if (x <= 1.85e+152) {
		tmp = fma(z, (z * 0.5), 1.0) - z;
	} else {
		tmp = x * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.15e+19)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 1.85e+152)
		tmp = Float64(fma(z, Float64(z * 0.5), 1.0) - z);
	else
		tmp = Float64(x * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.15e+19], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.85e+152], N[(N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15e19

   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 36.8

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

   5. Simplified 36.8%

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

   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 11.5

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

   8. Simplified 11.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   if -3.15e19 < x < 1.84999999999999998e152

   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 62.3

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

   5. Simplified 62.3%

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

   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 34.1

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

   8. Simplified 34.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)} \]
   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. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 34.1

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

   10. Applied egg-rr 34.1%

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

   if 1.84999999999999998e152 < 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

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

      1. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.9

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

   8. Simplified 89.9%

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

   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. lower-fma.f64 87.7

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

   11. Simplified 87.7%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 87.7

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

   14. Simplified 87.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.15e+19)
   (* (* z z) 0.5)
   (if (<= x 1.85e+152) (fma z (fma z 0.5 -1.0) 1.0) (* x (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = (z * z) * 0.5;
	} else if (x <= 1.85e+152) {
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	} else {
		tmp = x * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.15e+19)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 1.85e+152)
		tmp = fma(z, fma(z, 0.5, -1.0), 1.0);
	else
		tmp = Float64(x * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.15e+19], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.85e+152], N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15e19

   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 36.8

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

   5. Simplified 36.8%

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

   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 11.5

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

   8. Simplified 11.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   if -3.15e19 < x < 1.84999999999999998e152

   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 62.3

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

   5. Simplified 62.3%

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

   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 34.1

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

   8. Simplified 34.1%

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

   if 1.84999999999999998e152 < 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

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

      1. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.9

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

   8. Simplified 89.9%

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

   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. lower-fma.f64 87.7

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

   11. Simplified 87.7%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 87.7

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

   14. Simplified 87.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 41.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.15e+19)
   (* (* z z) 0.5)
   (if (<= x 1.85e+152) (fma z (* z 0.5) 1.0) (* x (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = (z * z) * 0.5;
	} else if (x <= 1.85e+152) {
		tmp = fma(z, (z * 0.5), 1.0);
	} else {
		tmp = x * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.15e+19)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 1.85e+152)
		tmp = fma(z, Float64(z * 0.5), 1.0);
	else
		tmp = Float64(x * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.15e+19], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.85e+152], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15e19

   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 36.8

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

   5. Simplified 36.8%

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

   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 11.5

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

   8. Simplified 11.5%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 28.8

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

   11. Simplified 28.8%

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

   if -3.15e19 < x < 1.84999999999999998e152

   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 62.3

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

   5. Simplified 62.3%

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

   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 34.1

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

   8. Simplified 34.1%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 33.9

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

   11. Simplified 33.9%

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

   if 1.84999999999999998e152 < 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

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

      1. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 89.9

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

   8. Simplified 89.9%

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

   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. lower-fma.f64 87.7

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

   11. Simplified 87.7%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 87.7

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

   14. Simplified 87.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 40.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e+152)
   (* (* z z) 0.5)
   (if (<= z 1.25e+15) (fma x (* x 0.5) 1.0) (* x (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+152) {
		tmp = (z * z) * 0.5;
	} else if (z <= 1.25e+15) {
		tmp = fma(x, (x * 0.5), 1.0);
	} else {
		tmp = x * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e+152)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (z <= 1.25e+15)
		tmp = fma(x, Float64(x * 0.5), 1.0);
	else
		tmp = Float64(x * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e+152], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[z, 1.25e+15], N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7000000000000001e152

   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 96.7

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

   5. Simplified 96.7%

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

   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 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 93.8

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

   11. Simplified 93.8%

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

   if -1.7000000000000001e152 < z < 1.25e15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 60.5

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

   8. Simplified 60.5%

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

   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. lower-fma.f64 34.8

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

   11. Simplified 34.8%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 33.9

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

   14. Simplified 33.9%

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

   if 1.25e15 < 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

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

      1. lower--.f64 79.7

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

   5. Simplified 79.7%

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

   6. Taylor expanded in z around 0

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

      1. lower-exp.f64 41.2

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

   8. Simplified 41.2%

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

   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. lower-fma.f64 18.9

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

   11. Simplified 18.9%

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

   12. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 29.9

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

   14. Simplified 29.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 14.6% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 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 = x + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 77.7

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

5. Simplified 77.7%

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

6. Taylor expanded in z around 0

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

   1. lower-exp.f64 50.5

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

8. Simplified 50.5%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 14.4

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

11. Simplified 14.4%

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

12. Final simplification 14.4%

    \[\leadsto x + 1 \]
13. Add Preprocessing

Alternative 23: 14.4% 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 51.0

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

5. Simplified 51.0%

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

6. Taylor expanded in z around 0

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

8. Simplified 13.8%

   \[\leadsto \color{blue}{1} \]
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 2024219 
(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)))