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

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 19

Speedup: 1.0×

• 58.6% 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: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4 - \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)))
   (if (<= t_0 -5e+146)
     (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))
     (if (<= t_0 -1e+14)
       (* -0.16666666666666666 (* z (* z z)))
       (if (<= t_0 5e+17)
         (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
         (fma
          (* z (fma 0.25 (* z z) -1.0))
          (/ (+ 2.0 (/ (- -4.0 (/ (+ (/ 16.0 z) -8.0) z)) z)) z)
          1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double tmp;
	if (t_0 <= -5e+146) {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else if (t_0 <= -1e+14) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (t_0 <= 5e+17) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 + ((-4.0 - (((16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_0 <= -5e+146)
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	elseif (t_0 <= -1e+14)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (t_0 <= 5e+17)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 + Float64(Float64(-4.0 - Float64(Float64(Float64(16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+146], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+14], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+17], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(N[(-4.0 - N[(N[(N[(16.0 / z), $MachinePrecision] + -8.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -4.9999999999999999e146

   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. neg-lowering-neg.f64 60.6

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

   5. Simplified 60.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 2.1

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

   8. Simplified 2.1%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 1.3%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 51.3%

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

   if -4.9999999999999999e146 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1e14

   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. neg-lowering-neg.f64 43.8

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

   5. Simplified 43.8%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 49.3

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 49.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1e14 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5e17

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 86.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 80.4

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

   8. Simplified 80.4%

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

   if 5e17 < (-.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. neg-lowering-neg.f64 35.8

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

   5. Simplified 35.8%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 23.9

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

   8. Simplified 23.9%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 32.0

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

   10. Applied egg-rr 32.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{\left(2 + \frac{8}{{z}^{2}}\right) - \left(4 \cdot \frac{1}{z} + 16 \cdot \frac{1}{{z}^{3}}\right)}{z}}, 1\right) \]

   13. Simplified 55.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{2 + \frac{-4 - \frac{\frac{16}{z} + -8}{z}}{z}}{z}}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq -1 \cdot 10^{+14}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4 - \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)))
   (if (<= t_0 -5e+146)
     (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))
     (if (<= t_0 -1e+14)
       (* -0.16666666666666666 (* z (* z z)))
       (if (<= t_0 5e+17)
         (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
         (fma (* z (fma 0.25 (* z z) -1.0)) (/ (+ 2.0 (/ -4.0 z)) z) 1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double tmp;
	if (t_0 <= -5e+146) {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else if (t_0 <= -1e+14) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (t_0 <= 5e+17) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 + (-4.0 / z)) / z), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_0 <= -5e+146)
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	elseif (t_0 <= -1e+14)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (t_0 <= 5e+17)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 + Float64(-4.0 / z)) / z), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+146], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+14], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+17], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(-4.0 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -4.9999999999999999e146

   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. neg-lowering-neg.f64 60.6

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

   5. Simplified 60.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 2.1

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

   8. Simplified 2.1%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 1.3%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 51.3%

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

   if -4.9999999999999999e146 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1e14

   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. neg-lowering-neg.f64 43.8

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

   5. Simplified 43.8%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 49.3

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 49.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1e14 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5e17

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 86.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 80.4

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

   8. Simplified 80.4%

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

   if 5e17 < (-.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. neg-lowering-neg.f64 35.8

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

   5. Simplified 35.8%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 23.9

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

   8. Simplified 23.9%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 32.0

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

   10. Applied egg-rr 32.0%

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

   11. Taylor expanded in z around inf

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

       1. /-lowering-/.f64 N/A

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

       2. sub-neg N/A

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

       3. +-lowering-+.f64 N/A

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

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

       6. distribute-neg-frac N/A

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

       7. /-lowering-/.f64 N/A

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

       8. metadata-eval 50.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{2 + \frac{\color{blue}{-4}}{z}}{z}, 1\right) \]

   13. Simplified 50.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq -1 \cdot 10^{+14}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+69}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -1e+14) (exp x) (if (<= t_0 1e+69) (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 <= -1e+14) {
		tmp = exp(x);
	} else if (t_0 <= 1e+69) {
		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 <= (-1d+14)) then
        tmp = exp(x)
    else if (t_0 <= 1d+69) 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 <= -1e+14) {
		tmp = Math.exp(x);
	} else if (t_0 <= 1e+69) {
		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 <= -1e+14:
		tmp = math.exp(x)
	elif t_0 <= 1e+69:
		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 <= -1e+14)
		tmp = exp(x);
	elseif (t_0 <= 1e+69)
		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 <= -1e+14)
		tmp = exp(x);
	elseif (t_0 <= 1e+69)
		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, -1e+14], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e+69], 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))) < -1e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.7%

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

   if -1e14 < (+.f64 x (*.f64 y (log.f64 y))) < 1.0000000000000001e69

   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. neg-lowering-neg.f64 78.6

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

   5. Simplified 78.6%

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

   if 1.0000000000000001e69 < (+.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. *-lowering-*.f64 N/A

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

      6. log-lowering-log.f64 82.8

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

   5. Simplified 82.8%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. pow-lowering-pow.f64 82.8

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

   7. Applied egg-rr 82.8%

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

Alternative 5: 33.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \left(z \cdot z\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+125}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* (* z z) 0.5)))
   (if (<= t_0 -1e+14) t_1 (if (<= t_0 4e+125) (+ 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 <= -1e+14) {
		tmp = t_1;
	} else if (t_0 <= 4e+125) {
		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 <= (-1d+14)) then
        tmp = t_1
    else if (t_0 <= 4d+125) 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 <= -1e+14) {
		tmp = t_1;
	} else if (t_0 <= 4e+125) {
		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 <= -1e+14:
		tmp = t_1
	elif t_0 <= 4e+125:
		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 <= -1e+14)
		tmp = t_1;
	elseif (t_0 <= 4e+125)
		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 <= -1e+14)
		tmp = t_1;
	elseif (t_0 <= 4e+125)
		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, -1e+14], t$95$1, If[LessEqual[t$95$0, 4e+125], 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) < -1e14 or 3.9999999999999997e125 < (-.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. neg-lowering-neg.f64 46.2

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

   5. Simplified 46.2%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 20.4

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

   8. Simplified 20.4%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 26.4

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

   11. Simplified 26.4%

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

   if -1e14 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 3.9999999999999997e125

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.3%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 37.4

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

   8. Simplified 37.4%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 4 \cdot 10^{+125}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x (* y (log y))) -1e+14) (* (* z z) 0.5) (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + (y * log(y))) <= -1e+14) {
		tmp = (z * z) * 0.5;
	} else {
		tmp = fma(z, (z * 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(y * log(y))) <= -1e+14)
		tmp = Float64(Float64(z * z) * 0.5);
	else
		tmp = fma(z, Float64(z * 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+14], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -1e14

   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. neg-lowering-neg.f64 37.9

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

   5. Simplified 37.9%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 8.8

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

   8. Simplified 8.8%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 34.4

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

   11. Simplified 34.4%

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

   if -1e14 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 48.2

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

   5. Simplified 48.2%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 28.6

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

   8. Simplified 28.6%

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

   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. *-lowering-*.f64 28.3

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

   11. Simplified 28.3%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+183}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.02e+121)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= z 1.95e+183) (exp x) (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.02e+121) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (z <= 1.95e+183) {
		tmp = exp(x);
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.02e+121)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (z <= 1.95e+183)
		tmp = exp(x);
	else
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.02e+121], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+183], N[Exp[x], $MachinePrecision], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02000000000000005e121

   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. neg-lowering-neg.f64 97.3

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

   5. Simplified 97.3%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 97.3

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

   8. Simplified 97.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 97.3

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 97.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1.02000000000000005e121 < z < 1.9499999999999999e183

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.2%

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

   if 1.9499999999999999e183 < 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. neg-lowering-neg.f64 87.7

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

   5. Simplified 87.7%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 13.9

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

   8. Simplified 13.9%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 87.7%

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

Alternative 8: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 82000000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 82000000.0) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 82000000.0) {
		tmp = exp(x);
	} 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 <= 82000000.0d0) then
        tmp = exp(x)
    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 <= 82000000.0) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 82000000.0:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 82000000.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 82000000.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 82000000.0], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 72.3%

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

   if 8.2e7 < 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. *-lowering-*.f64 N/A

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

      6. log-lowering-log.f64 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. pow-lowering-pow.f64 81.4

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

   7. Applied egg-rr 81.4%

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

Alternative 9: 51.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+73}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.25, -0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+73)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= z 8.2e+45)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (if (<= z 2e+183)
       (fma (* z (fma 0.25 (* z z) -1.0)) (fma z (fma z 0.25 -0.5) 1.0) 1.0)
       (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+73) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (z <= 8.2e+45) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (z <= 2e+183) {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), fma(z, fma(z, 0.25, -0.5), 1.0), 1.0);
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+73)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (z <= 8.2e+45)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	elseif (z <= 2e+183)
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), fma(z, fma(z, 0.25, -0.5), 1.0), 1.0);
	else
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+73], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.2e+45], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2e+183], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(z * N[(z * 0.25 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.50000000000000005e73

   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. neg-lowering-neg.f64 89.1

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

   5. Simplified 89.1%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 80.9

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

   8. Simplified 80.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 80.9

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 80.9%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1.50000000000000005e73 < z < 8.20000000000000025e45

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 35.4

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

   8. Simplified 35.4%

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

   if 8.20000000000000025e45 < z < 1.99999999999999989e183

   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. neg-lowering-neg.f64 44.2

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

   5. Simplified 44.2%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 20.0

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

   8. Simplified 20.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 28.9

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

   10. Applied egg-rr 28.9%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 48.0

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

   13. Simplified 48.0%

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

   if 1.99999999999999989e183 < 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. neg-lowering-neg.f64 87.7

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

   5. Simplified 87.7%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 13.9

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

   8. Simplified 13.9%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 87.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+73}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.25, -0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+73}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+73)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= z 1.15e+183)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+73) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (z <= 1.15e+183) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+73)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (z <= 1.15e+183)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+73], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+183], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000005e73

   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. neg-lowering-neg.f64 89.1

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

   5. Simplified 89.1%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 80.9

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

   8. Simplified 80.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 80.9

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 80.9%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1.50000000000000005e73 < z < 1.1499999999999999e183

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 34.3

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

   8. Simplified 34.3%

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

   if 1.1499999999999999e183 < 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. neg-lowering-neg.f64 87.7

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

   5. Simplified 87.7%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 13.9

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

   8. Simplified 13.9%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 87.7%

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

Alternative 11: 48.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -35000000000.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 4.6e+87)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 4.6e+87) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -35000000000.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 4.6e+87)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -35000000000.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e+87], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e10

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 10.3

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

   8. Simplified 10.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 36.6

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 36.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -3.5e10 < x < 4.6000000000000003e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 33.5

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

   8. Simplified 33.5%

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

   if 4.6000000000000003e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 95.9

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

   8. Simplified 95.9%

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

Alternative 12: 48.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -35000000000.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 3.05e+87)
     (fma z (* -0.16666666666666666 (* z z)) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 3.05e+87) {
		tmp = fma(z, (-0.16666666666666666 * (z * z)), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -35000000000.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 3.05e+87)
		tmp = fma(z, Float64(-0.16666666666666666 * Float64(z * z)), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -35000000000.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.05e+87], N[(z * N[(-0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e10

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 10.3

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

   8. Simplified 10.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 36.6

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 36.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -3.5e10 < x < 3.0499999999999999e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 33.5

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

   8. Simplified 33.5%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 33.1

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

   11. Simplified 33.1%

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

   if 3.0499999999999999e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
   7. 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. accelerator-lowering-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. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 95.9

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

   8. Simplified 95.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2500000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2500000000000.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 3.2e+122)
     (fma z (* -0.16666666666666666 (* z z)) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2500000000000.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 3.2e+122) {
		tmp = fma(z, (-0.16666666666666666 * (z * z)), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2500000000000.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 3.2e+122)
		tmp = fma(z, Float64(-0.16666666666666666 * Float64(z * z)), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2500000000000.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+122], N[(z * N[(-0.16666666666666666 * N[(z * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e12

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 10.3

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

   8. Simplified 10.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 36.6

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 36.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -2.5e12 < x < 3.20000000000000012e122

   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. neg-lowering-neg.f64 54.0

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

   5. Simplified 54.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 33.2

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

   8. Simplified 33.2%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 32.7

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

   11. Simplified 32.7%

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

   if 3.20000000000000012e122 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, 0.5, 1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -35000000000.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 4.6e+87)
     (fma (* z z) 0.5 (- 1.0 z))
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 4.6e+87) {
		tmp = fma((z * z), 0.5, (1.0 - z));
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -35000000000.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 4.6e+87)
		tmp = fma(Float64(z * z), 0.5, Float64(1.0 - z));
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -35000000000.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e+87], N[(N[(z * z), $MachinePrecision] * 0.5 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e10

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 10.3

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

   8. Simplified 10.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 36.6

          \[\leadsto -0.16666666666666666 \cdot \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

   11. Simplified 36.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -3.5e10 < x < 4.6000000000000003e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 31.3

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

   8. Simplified 31.3%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. --lowering--.f64 31.3

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

   10. Applied egg-rr 31.3%

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

   if 4.6000000000000003e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 82.0

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

   8. Simplified 82.0%

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

Alternative 15: 41.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -860000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, 0.5, 1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -860000000000.0)
   (* (* z z) 0.5)
   (if (<= x 4.6e+87)
     (fma (* z z) 0.5 (- 1.0 z))
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -860000000000.0) {
		tmp = (z * z) * 0.5;
	} else if (x <= 4.6e+87) {
		tmp = fma((z * z), 0.5, (1.0 - z));
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -860000000000.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 4.6e+87)
		tmp = fma(Float64(z * z), 0.5, Float64(1.0 - z));
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -860000000000.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.6e+87], N[(N[(z * z), $MachinePrecision] * 0.5 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.6e11

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 15.1

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

   8. Simplified 15.1%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 34.1

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

   11. Simplified 34.1%

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

   if -8.6e11 < x < 4.6000000000000003e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 31.3

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

   8. Simplified 31.3%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. --lowering--.f64 31.3

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

   10. Applied egg-rr 31.3%

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

   if 4.6000000000000003e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 82.0

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

   8. Simplified 82.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -860000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, 0.5, 1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -48000000000.0)
   (* (* z z) 0.5)
   (if (<= x 4e+87) (fma z (fma 0.5 z -1.0) 1.0) (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -48000000000.0) {
		tmp = (z * z) * 0.5;
	} else if (x <= 4e+87) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -48000000000.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 4e+87)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -48000000000.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4e+87], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e10

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 15.1

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

   8. Simplified 15.1%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 34.1

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

   11. Simplified 34.1%

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

   if -4.8e10 < x < 3.9999999999999998e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 31.3

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

   8. Simplified 31.3%

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

   if 3.9999999999999998e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 82.0

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

   8. Simplified 82.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -62000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -62000000000.0)
   (* (* z z) 0.5)
   (if (<= x 2.7e+87) (fma z (* z 0.5) 1.0) (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -62000000000.0) {
		tmp = (z * z) * 0.5;
	} else if (x <= 2.7e+87) {
		tmp = fma(z, (z * 0.5), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -62000000000.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (x <= 2.7e+87)
		tmp = fma(z, Float64(z * 0.5), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -62000000000.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.7e+87], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e10

   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. neg-lowering-neg.f64 33.6

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

   5. Simplified 33.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 15.1

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

   8. Simplified 15.1%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 34.1

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

   11. Simplified 34.1%

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

   if -6.2e10 < x < 2.70000000000000007e87

   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. neg-lowering-neg.f64 55.1

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

   5. Simplified 55.1%

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

   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. accelerator-lowering-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. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 31.3

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

   8. Simplified 31.3%

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

   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. *-lowering-*.f64 30.9

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

   11. Simplified 30.9%

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

   if 2.70000000000000007e87 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 82.0

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

   8. Simplified 82.0%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62000000000:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 14.7% 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 x around inf

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

5. Simplified 53.6%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 12.4

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

8. Simplified 12.4%

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

9. Final simplification 12.4%

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

Alternative 19: 14.5% 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 x around inf

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

5. Simplified 53.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 12.0%

   \[\leadsto \color{blue}{1} \]
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 2024204 
(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)))