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

Percentage Accurate: 100.0% → 100.0%

Time: 11.4s

Alternatives: 20

Speedup: 1.0×

• 57.6% 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: 51.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)\\ t_2 := z \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \frac{-6}{z \cdot z}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+212}:\\ \;\;\;\;z \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666 + \frac{-1}{z \cdot z}, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z))
        (t_1 (* z (fma z -0.16666666666666666 0.5)))
        (t_2 (* z (fma t_1 t_1 -1.0))))
   (if (<= t_0 0.5)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_0 2e+80)
       (fma t_2 (/ -6.0 (* z z)) 1.0)
       (if (<= t_0 2e+212)
         (* z (* z (fma z (+ -0.16666666666666666 (/ -1.0 (* z z))) 0.5)))
         (fma t_2 (/ 1.0 (fma z 0.5 1.0)) 1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = z * fma(z, -0.16666666666666666, 0.5);
	double t_2 = z * fma(t_1, t_1, -1.0);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 2e+80) {
		tmp = fma(t_2, (-6.0 / (z * z)), 1.0);
	} else if (t_0 <= 2e+212) {
		tmp = z * (z * fma(z, (-0.16666666666666666 + (-1.0 / (z * z))), 0.5));
	} else {
		tmp = fma(t_2, (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(z * fma(z, -0.16666666666666666, 0.5))
	t_2 = Float64(z * fma(t_1, t_1, -1.0))
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 2e+80)
		tmp = fma(t_2, Float64(-6.0 / Float64(z * z)), 1.0);
	elseif (t_0 <= 2e+212)
		tmp = Float64(z * Float64(z * fma(z, Float64(-0.16666666666666666 + Float64(-1.0 / Float64(z * z))), 0.5)));
	else
		tmp = fma(t_2, Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+80], N[(t$95$2 * N[(-6.0 / N[(z * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+212], N[(z * N[(z * N[(z * N[(-0.16666666666666666 + N[(-1.0 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 0.5

   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 67.4

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

   5. Simplified 67.4%

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

   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 30.8

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

   8. Simplified 30.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 30.8%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 65.8

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

   13. Simplified 65.8%

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

   if 0.5 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2e80

   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 31.3

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

   5. Simplified 31.3%

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

   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 3.4

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

   8. Simplified 3.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 15.1%

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

   11. Taylor expanded in z around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 27.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right) \cdot z, \frac{-6}{\color{blue}{z \cdot z}}, 1\right) \]

   13. Simplified 27.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right) \cdot z, \color{blue}{\frac{-6}{z \cdot z}}, 1\right) \]

   if 2e80 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.9999999999999998e212

   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 45.3

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

   5. Simplified 45.3%

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

   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 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. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 33.2%

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

   11. Taylor expanded in z around inf

       \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*r* N/A

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

       15. rgt-mult-inverse N/A

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

       16. metadata-eval N/A

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

   13. Simplified 55.7%

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

   if 1.9999999999999998e212 < (-.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 44.2

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

   5. Simplified 44.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   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 38.9

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

   8. Simplified 38.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 5.4%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 56.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right) \cdot z, \frac{1}{\mathsf{fma}\left(z, \color{blue}{0.5}, 1\right)}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{-6}{z \cdot z}, 1\right)\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+212}:\\ \;\;\;\;z \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666 + \frac{-1}{z \cdot z}, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+106}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+32}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -2e+106) (exp x) (if (<= t_0 2e+32) (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 <= -2e+106) {
		tmp = exp(x);
	} else if (t_0 <= 2e+32) {
		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 <= (-2d+106)) then
        tmp = exp(x)
    else if (t_0 <= 2d+32) 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 <= -2e+106) {
		tmp = Math.exp(x);
	} else if (t_0 <= 2e+32) {
		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 <= -2e+106:
		tmp = math.exp(x)
	elif t_0 <= 2e+32:
		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 <= -2e+106)
		tmp = exp(x);
	elseif (t_0 <= 2e+32)
		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 <= -2e+106)
		tmp = exp(x);
	elseif (t_0 <= 2e+32)
		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, -2e+106], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2e+32], 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))) < -2.00000000000000018e106

   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 94.4%

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

   if -2.00000000000000018e106 < (+.f64 x (*.f64 y (log.f64 y))) < 2.00000000000000011e32

   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 86.3

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

   5. Simplified 86.3%

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

   if 2.00000000000000011e32 < (+.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 73.4

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

   5. Simplified 73.4%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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 73.4

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

   7. Applied egg-rr 73.4%

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

Alternative 4: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)\\ t_2 := z \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \frac{-6 + \frac{-18}{z}}{z \cdot z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z))
        (t_1 (* z (fma z -0.16666666666666666 0.5)))
        (t_2 (* z (fma t_1 t_1 -1.0))))
   (if (<= t_0 0.5)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_0 1e+156)
       (fma t_2 (/ (+ -6.0 (/ -18.0 z)) (* z z)) 1.0)
       (fma t_2 (/ 1.0 (fma z 0.5 1.0)) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = z * fma(z, -0.16666666666666666, 0.5);
	double t_2 = z * fma(t_1, t_1, -1.0);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_0 <= 1e+156) {
		tmp = fma(t_2, ((-6.0 + (-18.0 / z)) / (z * z)), 1.0);
	} else {
		tmp = fma(t_2, (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(z * fma(z, -0.16666666666666666, 0.5))
	t_2 = Float64(z * fma(t_1, t_1, -1.0))
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_0 <= 1e+156)
		tmp = fma(t_2, Float64(Float64(-6.0 + Float64(-18.0 / z)) / Float64(z * z)), 1.0);
	else
		tmp = fma(t_2, Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+156], N[(t$95$2 * N[(N[(-6.0 + N[(-18.0 / z), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 0.5

   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 67.4

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

   5. Simplified 67.4%

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

   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 30.8

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

   8. Simplified 30.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 30.8%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 65.8

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

   13. Simplified 65.8%

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

   if 0.5 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 9.9999999999999998e155

   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 37.8

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

   5. Simplified 37.8%

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

   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 18.8

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

   8. Simplified 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 28.1%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \color{blue}{-1 \cdot \frac{6 + 18 \cdot \frac{1}{z}}{{z}^{2}}}, 1\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \color{blue}{\frac{-1 \cdot \left(6 + 18 \cdot \frac{1}{z}\right)}{{z}^{2}}}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \color{blue}{\frac{-1 \cdot \left(6 + 18 \cdot \frac{1}{z}\right)}{{z}^{2}}}, 1\right) \]

       3. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{\color{blue}{\mathsf{neg}\left(\left(6 + 18 \cdot \frac{1}{z}\right)\right)}}{{z}^{2}}, 1\right) \]

       4. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{\color{blue}{\left(\mathsf{neg}\left(6\right)\right) + \left(\mathsf{neg}\left(18 \cdot \frac{1}{z}\right)\right)}}{{z}^{2}}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{\color{blue}{-6} + \left(\mathsf{neg}\left(18 \cdot \frac{1}{z}\right)\right)}{{z}^{2}}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{\color{blue}{-6 + \left(\mathsf{neg}\left(18 \cdot \frac{1}{z}\right)\right)}}{{z}^{2}}, 1\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \left(\mathsf{neg}\left(\color{blue}{\frac{18 \cdot 1}{z}}\right)\right)}{{z}^{2}}, 1\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \left(\mathsf{neg}\left(\frac{\color{blue}{18}}{z}\right)\right)}{{z}^{2}}, 1\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \color{blue}{\frac{\mathsf{neg}\left(18\right)}{z}}}{{z}^{2}}, 1\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \color{blue}{\frac{\mathsf{neg}\left(18\right)}{z}}}{{z}^{2}}, 1\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \frac{\color{blue}{-18}}{z}}{{z}^{2}}, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), z \cdot \mathsf{fma}\left(z, \frac{-1}{6}, \frac{1}{2}\right), -1\right) \cdot z, \frac{-6 + \frac{-18}{z}}{\color{blue}{z \cdot z}}, 1\right) \]

       13. *-lowering-*.f64 65.1

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right) \cdot z, \frac{-6 + \frac{-18}{z}}{\color{blue}{z \cdot z}}, 1\right) \]

   13. Simplified 65.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right) \cdot z, \color{blue}{\frac{-6 + \frac{-18}{z}}{z \cdot z}}, 1\right) \]

   if 9.9999999999999998e155 < (-.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 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   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 37.3

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

   8. Simplified 37.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 7.3%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 52.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{-6 + \frac{-18}{z}}{z \cdot z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)\\ t_1 := \left(x + y \cdot \log y\right) - z\\ \mathbf{if}\;t\_1 \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, \frac{1}{z \cdot \left(z \cdot z\right)} + \left(-0.16666666666666666 + \frac{-1}{z \cdot z}\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t\_0, t\_0, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (fma z -0.16666666666666666 0.5)))
        (t_1 (- (+ x (* y (log y))) z)))
   (if (<= t_1 0.5)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= t_1 2e+49)
       (*
        (* z z)
        (fma
         z
         (+ (/ 1.0 (* z (* z z))) (+ -0.16666666666666666 (/ -1.0 (* z z))))
         0.5))
       (fma (* z (fma t_0 t_0 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * fma(z, -0.16666666666666666, 0.5);
	double t_1 = (x + (y * log(y))) - z;
	double tmp;
	if (t_1 <= 0.5) {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (t_1 <= 2e+49) {
		tmp = (z * z) * fma(z, ((1.0 / (z * (z * z))) + (-0.16666666666666666 + (-1.0 / (z * z)))), 0.5);
	} else {
		tmp = fma((z * fma(t_0, t_0, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * fma(z, -0.16666666666666666, 0.5))
	t_1 = Float64(Float64(x + Float64(y * log(y))) - z)
	tmp = 0.0
	if (t_1 <= 0.5)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (t_1 <= 2e+49)
		tmp = Float64(Float64(z * z) * fma(z, Float64(Float64(1.0 / Float64(z * Float64(z * z))) + Float64(-0.16666666666666666 + Float64(-1.0 / Float64(z * z)))), 0.5));
	else
		tmp = fma(Float64(z * fma(t_0, t_0, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, 0.5], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+49], N[(N[(z * z), $MachinePrecision] * N[(z * N[(N[(1.0 / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 + N[(-1.0 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 0.5

   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 67.4

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

   5. Simplified 67.4%

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

   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 30.8

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

   8. Simplified 30.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 30.8%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 65.8

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

   13. Simplified 65.8%

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

   if 0.5 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.99999999999999989e49

   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 19.3

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

   5. Simplified 19.3%

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

   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 3.2

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

   8. Simplified 3.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 3.2%

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

   11. Taylor expanded in z around inf

       \[\leadsto \color{blue}{{z}^{3} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot z\right)} \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{z}^{2}} \cdot z\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot \left(z \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {z}^{2} \cdot \color{blue}{\left(\left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \cdot z\right)} \]

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

          \[\leadsto \color{blue}{{z}^{2} \cdot \left(\left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \cdot z\right)} \]

       6. unpow2 N/A

          \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \left(\left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \cdot z\right) \]

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

          \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \left(\left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) \cdot z\right) \]

       8. *-commutative N/A

          \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\left(z \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{z} + \frac{1}{{z}^{3}}\right) - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)\right)} \]

       9. associate--l+ N/A

          \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{z} + \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)\right)}\right) \]

       10. +-commutative N/A

           \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{\left(\left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{z}\right)}\right) \]

       11. distribute-lft-in N/A

           \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\left(z \cdot \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + z \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + z \cdot \color{blue}{\left(\frac{1}{z} \cdot \frac{1}{2}\right)}\right) \]

       13. associate-*r* N/A

           \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + \color{blue}{\left(z \cdot \frac{1}{z}\right) \cdot \frac{1}{2}}\right) \]

       14. rgt-mult-inverse N/A

           \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + \color{blue}{1} \cdot \frac{1}{2}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(z \cdot z\right) \cdot \left(z \cdot \left(\frac{1}{{z}^{3}} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right) + \color{blue}{\frac{1}{2}}\right) \]

   13. Simplified 43.3%

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

   if 1.99999999999999989e49 < (-.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 44.5

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

   5. Simplified 44.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 33.7

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

   8. Simplified 33.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 14.9%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 49.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, \frac{1}{z \cdot \left(z \cdot z\right)} + \left(-0.16666666666666666 + \frac{-1}{z \cdot z}\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := 0.5 \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+96}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* 0.5 (* z z))))
   (if (<= t_0 -2e+25) t_1 (if (<= t_0 5e+96) (+ 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 = 0.5 * (z * z);
	double tmp;
	if (t_0 <= -2e+25) {
		tmp = t_1;
	} else if (t_0 <= 5e+96) {
		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 = 0.5d0 * (z * z)
    if (t_0 <= (-2d+25)) then
        tmp = t_1
    else if (t_0 <= 5d+96) 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 = 0.5 * (z * z);
	double tmp;
	if (t_0 <= -2e+25) {
		tmp = t_1;
	} else if (t_0 <= 5e+96) {
		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 = 0.5 * (z * z)
	tmp = 0
	if t_0 <= -2e+25:
		tmp = t_1
	elif t_0 <= 5e+96:
		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(0.5 * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2e+25)
		tmp = t_1;
	elseif (t_0 <= 5e+96)
		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 = 0.5 * (z * z);
	tmp = 0.0;
	if (t_0 <= -2e+25)
		tmp = t_1;
	elseif (t_0 <= 5e+96)
		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[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+25], t$95$1, If[LessEqual[t$95$0, 5e+96], 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) < -2.00000000000000018e25 or 5.0000000000000004e96 < (-.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 49.1

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

   5. Simplified 49.1%

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

   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 24.3

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

   8. Simplified 24.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   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 29.6

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

   11. Simplified 29.6%

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

   if -2.00000000000000018e25 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5.0000000000000004e96

   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 68.0%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 49.8

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

   8. Simplified 49.8%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -2 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 5 \cdot 10^{+96}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 0:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp (- (+ x (* y (log y))) z)) 0.0)
   (* 0.5 (* z z))
   (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(((x + (y * log(y))) - z)) <= 0.0) {
		tmp = 0.5 * (z * z);
	} else {
		tmp = fma(z, (z * 0.5), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)) < 0.0

   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 56.3

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

   5. Simplified 56.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 2.2

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

   8. Simplified 2.2%

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

   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 18.0

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

   11. Simplified 18.0%

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

   if 0.0 < (exp.f64 (-.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 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   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 40.8

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

   8. Simplified 40.8%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.f64 40.8

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

   11. Simplified 40.8%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 0:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+106}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e+106)
   (exp x)
   (if (<= x 3.4e-5) (exp (- (* y (log y)) z)) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+106) {
		tmp = exp(x);
	} else if (x <= 3.4e-5) {
		tmp = exp(((y * log(y)) - z));
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.7d+106)) then
        tmp = exp(x)
    else if (x <= 3.4d-5) then
        tmp = exp(((y * log(y)) - z))
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+106) {
		tmp = Math.exp(x);
	} else if (x <= 3.4e-5) {
		tmp = Math.exp(((y * Math.log(y)) - z));
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.7e+106:
		tmp = math.exp(x)
	elif x <= 3.4e-5:
		tmp = math.exp(((y * math.log(y)) - z))
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e+106)
		tmp = exp(x);
	elseif (x <= 3.4e-5)
		tmp = exp(Float64(Float64(y * log(y)) - z));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.7e+106)
		tmp = exp(x);
	elseif (x <= 3.4e-5)
		tmp = exp(((y * log(y)) - z));
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e+106], N[Exp[x], $MachinePrecision], If[LessEqual[x, 3.4e-5], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], N[Exp[x], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999997e106 or 3.4e-5 < 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 87.8%

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

   if -1.69999999999999997e106 < x < 3.4e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. log-lowering-log.f64 96.8

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

   5. Simplified 96.8%

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

Alternative 9: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 10^{+72}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666 + \frac{-1}{z \cdot z}, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 1e+72)
   (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))
   (* z (* z (fma z (+ -0.16666666666666666 (/ -1.0 (* z z))) 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= 1e+72) {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = z * (z * fma(z, (-0.16666666666666666 + (-1.0 / (z * z))), 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * log(y))) - z) <= 1e+72)
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	else
		tmp = Float64(z * Float64(z * fma(z, Float64(-0.16666666666666666 + Float64(-1.0 / Float64(z * z))), 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 1e+72], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(z * N[(-0.16666666666666666 + N[(-1.0 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 9.99999999999999944e71

   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 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 26.1

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

   8. Simplified 26.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 26.1%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 55.1

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

   13. Simplified 55.1%

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

   if 9.99999999999999944e71 < (-.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 45.9

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

   5. Simplified 45.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 35.8

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

   8. Simplified 35.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 35.8%

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

   11. Taylor expanded in z around inf

       \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \left(\frac{1}{6} + \frac{1}{{z}^{2}}\right)\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*r* N/A

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

       15. rgt-mult-inverse N/A

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

       16. metadata-eval N/A

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

   13. Simplified 49.8%

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666 + \frac{-1}{z \cdot z}, 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t\_0, t\_0, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (fma z -0.16666666666666666 0.5))))
   (if (<= z -6.2e+53)
     (fma (* z (fma t_0 t_0 -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)
     (if (<= z 1.9e+115)
       (exp x)
       (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * fma(z, -0.16666666666666666, 0.5);
	double tmp;
	if (z <= -6.2e+53) {
		tmp = fma((z * fma(t_0, t_0, -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	} else if (z <= 1.9e+115) {
		tmp = exp(x);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * fma(z, -0.16666666666666666, 0.5))
	tmp = 0.0
	if (z <= -6.2e+53)
		tmp = fma(Float64(z * fma(t_0, t_0, -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	elseif (z <= 1.9e+115)
		tmp = exp(x);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.2e+53], N[(N[(z * N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.9e+115], N[Exp[x], $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.20000000000000038e53

   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 86.4

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

   5. Simplified 86.4%

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

   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 70.5

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

   8. Simplified 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 29.5%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 83.5%

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

   if -6.20000000000000038e53 < z < 1.9e115

   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 64.2%

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

   if 1.9e115 < 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 73.4

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

   5. Simplified 73.4%

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

   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 1.1

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

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.1%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 73.4

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

   13. Simplified 73.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 6.2e-6) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-6) {
		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 <= 6.2d-6) 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 <= 6.2e-6) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e-6:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e-6)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.2e-6)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e-6], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.1999999999999999e-6

   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 71.4%

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

   if 6.1999999999999999e-6 < 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 82.2

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

   5. Simplified 82.2%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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 82.2

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

   7. Applied egg-rr 82.2%

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

Alternative 12: 52.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot \mathsf{fma}\left(z \cdot z, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+77)
   (/
    (* (* z z) (fma (* z z) 0.027777777777777776 -0.25))
    (fma z -0.16666666666666666 -0.5))
   (if (<= z 1.9e+115)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+77) {
		tmp = ((z * z) * fma((z * z), 0.027777777777777776, -0.25)) / fma(z, -0.16666666666666666, -0.5);
	} else if (z <= 1.9e+115) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+77)
		tmp = Float64(Float64(Float64(z * z) * fma(Float64(z * z), 0.027777777777777776, -0.25)) / fma(z, -0.16666666666666666, -0.5));
	elseif (z <= 1.9e+115)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+77], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision]), $MachinePrecision] / N[(z * -0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e+115], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6000000000000002e77

   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 85.0

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

   5. Simplified 85.0%

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

   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 77.1

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

   8. Simplified 77.1%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-in N/A

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

       7. distribute-lft-in N/A

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

       8. associate-*l* N/A

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

       9. lft-mult-inverse N/A

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

       10. metadata-eval N/A

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

       11. distribute-lft-in N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. +-commutative N/A

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

       16. *-commutative N/A

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

       17. accelerator-lowering-fma.f64 77.1

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

   11. Simplified 77.1%

       \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. swap-sqr N/A

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

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

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

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

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

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right) \cdot \left(z \cdot z\right)}{z \cdot \frac{-1}{6} - \frac{1}{2}} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \left(z \cdot z\right)}{z \cdot \frac{-1}{6} - \frac{1}{2}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{\frac{-1}{4}}\right) \cdot \left(z \cdot z\right)}{z \cdot \frac{-1}{6} - \frac{1}{2}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{1}{36}, \frac{-1}{4}\right) \cdot \color{blue}{\left(z \cdot z\right)}}{z \cdot \frac{-1}{6} - \frac{1}{2}} \]

       14. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{1}{36}, \frac{-1}{4}\right) \cdot \left(z \cdot z\right)}{\color{blue}{z \cdot \frac{-1}{6} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

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

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{1}{36}, \frac{-1}{4}\right) \cdot \left(z \cdot z\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{-1}{6}, \mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

       16. metadata-eval 85.0

           \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, 0.027777777777777776, -0.25\right) \cdot \left(z \cdot z\right)}{\mathsf{fma}\left(z, -0.16666666666666666, \color{blue}{-0.5}\right)} \]

   13. Applied egg-rr 85.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, 0.027777777777777776, -0.25\right) \cdot \left(z \cdot z\right)}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}} \]

   if -2.6000000000000002e77 < z < 1.9e115

   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 63.1%

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

   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 39.4

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

   8. Simplified 39.4%

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

   if 1.9e115 < 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 73.4

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

   5. Simplified 73.4%

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

   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 1.1

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

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.1%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 73.4

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

   13. Simplified 73.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot \mathsf{fma}\left(z \cdot z, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+77)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= z 1.9e+115)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+77) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (z <= 1.9e+115) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+77)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (z <= 1.9e+115)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+77], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e+115], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6000000000000002e77

   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 85.0

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

   5. Simplified 85.0%

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

   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 77.1

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

   8. Simplified 77.1%

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

   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 77.1

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

   11. Simplified 77.1%

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

   if -2.6000000000000002e77 < z < 1.9e115

   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 63.1%

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

   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 39.4

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

   8. Simplified 39.4%

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

   if 1.9e115 < 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 73.4

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

   5. Simplified 73.4%

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

   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 1.1

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

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.1%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 73.4

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

   13. Simplified 73.4%

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

Alternative 14: 49.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+77}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e+77)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= z 3.3e+115)
     (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 <= -4.6e+77) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (z <= 3.3e+115) {
		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 <= -4.6e+77)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (z <= 3.3e+115)
		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, -4.6e+77], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e+115], 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 < -4.5999999999999999e77

   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 85.0

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

   5. Simplified 85.0%

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

   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 77.1

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

   8. Simplified 77.1%

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

   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 77.1

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

   11. Simplified 77.1%

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

   if -4.5999999999999999e77 < z < 3.30000000000000005e115

   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 63.1%

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

   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 39.4

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

   8. Simplified 39.4%

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

   if 3.30000000000000005e115 < 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 73.4

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

   5. Simplified 73.4%

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

   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 1.1

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

   8. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right)}^{3} + {1}^{3}}{\left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) + \left(1 \cdot 1 - \left(z \cdot \left(z \cdot \left(z \cdot \frac{-1}{6} + \frac{1}{2}\right) + -1\right)\right) \cdot 1\right)}}}} \]

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.1%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 55.9

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

   13. Simplified 55.9%

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

Alternative 15: 45.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+22)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.95e+60)
     (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 <= -1.5e+22) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.95e+60) {
		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 <= -1.5e+22)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.95e+60)
		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, -1.5e+22], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.95e+60], 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 < -1.5e22

   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 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 17.4

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

   8. Simplified 17.4%

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

   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 41.0

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

   11. Simplified 41.0%

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

   if -1.5e22 < x < 2.9500000000000001e60

   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 64.7

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

   5. Simplified 64.7%

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

   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 39.9

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

   8. Simplified 39.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot {z}^{2}}, 1\right) \]
   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 39.9

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

   11. Simplified 39.9%

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

   if 2.9500000000000001e60 < 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 92.0%

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

   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 80.8

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

   8. Simplified 80.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+22)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 1.25e+57)
     (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 <= -1.5e+22) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1.25e+57) {
		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 <= -1.5e+22)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 1.25e+57)
		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, -1.5e+22], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+57], 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 < -1.5e22

   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 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 17.4

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

   8. Simplified 17.4%

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

   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 41.0

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

   11. Simplified 41.0%

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

   if -1.5e22 < x < 1.24999999999999993e57

   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 64.7

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

   5. Simplified 64.7%

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

   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 39.9

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

   8. Simplified 39.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{-1}{6} \cdot {z}^{2}}, 1\right) \]
   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 39.9

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

   11. Simplified 39.9%

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

   if 1.24999999999999993e57 < 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 92.0%

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

   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 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(z, -0.16666666666666666 \cdot \left(z \cdot z\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.4e+22)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.95e+60) (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 <= -2.4e+22) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.95e+60) {
		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 <= -2.4e+22)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.95e+60)
		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, -2.4e+22], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.95e+60], 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 < -2.4e22

   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 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 17.4

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

   8. Simplified 17.4%

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

   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 41.0

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

   11. Simplified 41.0%

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

   if -2.4e22 < x < 2.9500000000000001e60

   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 64.7

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

   5. Simplified 64.7%

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

   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 38.9

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

   8. Simplified 38.9%

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

   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 38.9

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

   11. Simplified 38.9%

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

   if 2.9500000000000001e60 < 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 92.0%

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

   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 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.65e+22)
   (* 0.5 (* z z))
   (if (<= x 2.95e+60) (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 <= -1.65e+22) {
		tmp = 0.5 * (z * z);
	} else if (x <= 2.95e+60) {
		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 <= -1.65e+22)
		tmp = Float64(0.5 * Float64(z * z));
	elseif (x <= 2.95e+60)
		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, -1.65e+22], N[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.95e+60], 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 < -1.6499999999999999e22

   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 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. 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 16.4

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

   8. Simplified 16.4%

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

   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 35.0

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

   11. Simplified 35.0%

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

   if -1.6499999999999999e22 < x < 2.9500000000000001e60

   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 64.7

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

   5. Simplified 64.7%

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

   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 38.9

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

   8. Simplified 38.9%

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

   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 38.9

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

   11. Simplified 38.9%

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

   if 2.9500000000000001e60 < 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 92.0%

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

   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 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 14.5% 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 52.6%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 14.8

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

8. Simplified 14.8%

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

9. Final simplification 14.8%

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

Alternative 20: 14.3% 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 52.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 14.4%

   \[\leadsto \color{blue}{1} \]
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 2024198 
(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)))