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

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 21

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: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := z \cdot \left(z \cdot z\right)\\ t_2 := \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+160}:\\ \;\;\;\;\frac{-1}{t\_2}\\ \mathbf{elif}\;t\_0 \leq -20000000000000:\\ \;\;\;\;\frac{0.25 \cdot \left(z \cdot t\_1\right)}{t\_2}\\ \mathbf{elif}\;t\_0 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(t\_1, -0.004629629629629629, 0.125\right) \cdot \frac{1}{0.25 + -0.16666666666666666 \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)\right)}, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4 - \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z))
        (t_1 (* z (* z z)))
        (t_2 (fma z (fma z 0.5 -1.0) -1.0)))
   (if (<= t_0 -1e+160)
     (/ -1.0 t_2)
     (if (<= t_0 -20000000000000.0)
       (/ (* 0.25 (* z t_1)) t_2)
       (if (<= t_0 100000000.0)
         (fma
          z
          (fma
           z
           (*
            (fma t_1 -0.004629629629629629 0.125)
            (/
             1.0
             (+
              0.25
              (*
               -0.16666666666666666
               (* z (fma z -0.16666666666666666 -0.5))))))
           -1.0)
          1.0)
         (fma
          (* z (fma 0.25 (* z z) -1.0))
          (/ (+ 2.0 (/ (- -4.0 (/ (+ (/ 16.0 z) -8.0) z)) z)) z)
          1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = z * (z * z);
	double t_2 = fma(z, fma(z, 0.5, -1.0), -1.0);
	double tmp;
	if (t_0 <= -1e+160) {
		tmp = -1.0 / t_2;
	} else if (t_0 <= -20000000000000.0) {
		tmp = (0.25 * (z * t_1)) / t_2;
	} else if (t_0 <= 100000000.0) {
		tmp = fma(z, fma(z, (fma(t_1, -0.004629629629629629, 0.125) * (1.0 / (0.25 + (-0.16666666666666666 * (z * fma(z, -0.16666666666666666, -0.5)))))), -1.0), 1.0);
	} else {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 + ((-4.0 - (((16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(z * Float64(z * z))
	t_2 = fma(z, fma(z, 0.5, -1.0), -1.0)
	tmp = 0.0
	if (t_0 <= -1e+160)
		tmp = Float64(-1.0 / t_2);
	elseif (t_0 <= -20000000000000.0)
		tmp = Float64(Float64(0.25 * Float64(z * t_1)) / t_2);
	elseif (t_0 <= 100000000.0)
		tmp = fma(z, fma(z, Float64(fma(t_1, -0.004629629629629629, 0.125) * Float64(1.0 / Float64(0.25 + Float64(-0.16666666666666666 * Float64(z * fma(z, -0.16666666666666666, -0.5)))))), -1.0), 1.0);
	else
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 + Float64(Float64(-4.0 - Float64(Float64(Float64(16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+160], N[(-1.0 / t$95$2), $MachinePrecision], If[LessEqual[t$95$0, -20000000000000.0], N[(N[(0.25 * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 100000000.0], N[(z * N[(z * N[(N[(t$95$1 * -0.004629629629629629 + 0.125), $MachinePrecision] * N[(1.0 / N[(0.25 + N[(-0.16666666666666666 * N[(z * N[(z * -0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(N[(-4.0 - N[(N[(N[(16.0 / z), $MachinePrecision] + -8.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1.00000000000000001e160

   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-sub0 N/A

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

      3. --lowering--.f64 60.7

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

   5. Simplified 60.7%

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

   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. Step-by-step derivation

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 1.4%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 51.1%

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

   if -1.00000000000000001e160 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -2e13

   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-sub0 N/A

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

      3. --lowering--.f64 41.0

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

   5. Simplified 41.0%

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

   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 2.7

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

   8. Simplified 2.7%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 2.7%

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

   11. Taylor expanded in z around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 57.4

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

   13. Simplified 57.4%

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

   if -2e13 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1e8

   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-sub0 N/A

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

      3. --lowering--.f64 92.5

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

   5. Simplified 92.5%

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

   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 89.6

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

   8. Simplified 89.6%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

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

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

      6. unpow-prod-down N/A

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right) \cdot \frac{1}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right) \cdot \frac{1}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot \frac{1}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot \frac{1}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 89.6%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot \frac{1}{0.25 + -0.16666666666666666 \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)\right)}}, -1\right), 1\right) \]

   if 1e8 < (-.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-sub0 N/A

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

      3. --lowering--.f64 39.9

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

   5. Simplified 39.9%

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

   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 21.9

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

   8. Simplified 21.9%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 30.5

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

   10. Applied egg-rr 30.5%

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

   11. Taylor expanded in z around inf

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

   13. Simplified 51.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -1 \cdot 10^{+160}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq -20000000000000:\\ \;\;\;\;\frac{0.25 \cdot \left(z \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot \frac{1}{0.25 + -0.16666666666666666 \cdot \left(z \cdot \mathsf{fma}\left(z, -0.16666666666666666, -0.5\right)\right)}, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4 - \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+29}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -0.2) (exp x) (if (<= t_0 5e+29) (exp (- 0.0 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 <= -0.2) {
		tmp = exp(x);
	} else if (t_0 <= 5e+29) {
		tmp = exp((0.0 - 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 <= (-0.2d0)) then
        tmp = exp(x)
    else if (t_0 <= 5d+29) then
        tmp = exp((0.0d0 - 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 <= -0.2) {
		tmp = Math.exp(x);
	} else if (t_0 <= 5e+29) {
		tmp = Math.exp((0.0 - 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 <= -0.2:
		tmp = math.exp(x)
	elif t_0 <= 5e+29:
		tmp = math.exp((0.0 - 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 <= -0.2)
		tmp = exp(x);
	elseif (t_0 <= 5e+29)
		tmp = exp(Float64(0.0 - 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 <= -0.2)
		tmp = exp(x);
	elseif (t_0 <= 5e+29)
		tmp = exp((0.0 - 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, -0.2], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+29], N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.0%

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

   if -0.20000000000000001 < (+.f64 x (*.f64 y (log.f64 y))) < 5.0000000000000001e29

   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-sub0 N/A

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

      3. --lowering--.f64 94.8

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

   5. Simplified 94.8%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 94.8

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

   7. Applied egg-rr 94.8%

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

   if 5.0000000000000001e29 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. log-lowering-log.f64 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 68.6

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

   8. Simplified 68.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -0.2:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x + y \cdot \log y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (fma 0.5 (* z z) 0.0)))
   (if (<= t_0 -1e+20) t_1 (if (<= t_0 5e+72) (- 1.0 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = fma(0.5, (z * z), 0.0);
	double tmp;
	if (t_0 <= -1e+20) {
		tmp = t_1;
	} else if (t_0 <= 5e+72) {
		tmp = 1.0 - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = fma(0.5, Float64(z * z), 0.0)
	tmp = 0.0
	if (t_0 <= -1e+20)
		tmp = t_1;
	elseif (t_0 <= 5e+72)
		tmp = Float64(1.0 - z);
	else
		tmp = t_1;
	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[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+20], t$95$1, If[LessEqual[t$95$0, 5e+72], N[(1.0 - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1e20 or 4.99999999999999992e72 < (-.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-sub0 N/A

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

      3. --lowering--.f64 47.3

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

   5. Simplified 47.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 15.7

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

   8. Simplified 15.7%

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

   9. Taylor expanded in z around inf

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

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 23.8

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 23.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -1e20 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 4.99999999999999992e72

   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-sub0 N/A

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

      3. --lowering--.f64 71.0

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

   5. Simplified 71.0%

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

   6. Taylor expanded in z around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 62.8

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

   8. Simplified 62.8%

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

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+98}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 z))))
   (if (<= z -1.85e+72) t_0 (if (<= z 5.8e+98) (exp (fma y (log y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = exp((0.0 - z));
	double tmp;
	if (z <= -1.85e+72) {
		tmp = t_0;
	} else if (z <= 5.8e+98) {
		tmp = exp(fma(y, log(y), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = exp(Float64(0.0 - z))
	tmp = 0.0
	if (z <= -1.85e+72)
		tmp = t_0;
	elseif (z <= 5.8e+98)
		tmp = exp(fma(y, log(y), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.85e+72], t$95$0, If[LessEqual[z, 5.8e+98], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8500000000000001e72 or 5.8000000000000002e98 < 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-sub0 N/A

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

      3. --lowering--.f64 91.7

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

   5. Simplified 91.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 91.7

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

   7. Applied egg-rr 91.7%

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

   if -1.8500000000000001e72 < z < 5.8000000000000002e98

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. log-lowering-log.f64 95.7

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

   5. Simplified 95.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+72}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+98}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 8 \cdot 10^{+109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(0.5, z, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 8e+109) 1.0 (* z (fma 0.5 z -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= 8e+109) {
		tmp = 1.0;
	} else {
		tmp = z * fma(0.5, z, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * log(y))) - z) <= 8e+109)
		tmp = 1.0;
	else
		tmp = Float64(z * fma(0.5, z, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 8e+109], 1.0, N[(z * N[(0.5 * z + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 7.99999999999999985e109

   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 63.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 24.2%

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

   if 7.99999999999999985e109 < (-.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-sub0 N/A

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

      3. --lowering--.f64 45.5

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

   5. Simplified 45.5%

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

   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 27.4

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

   8. Simplified 27.4%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 37.7

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

   10. Applied egg-rr 37.7%

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

   11. Taylor expanded in z around inf

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

       1. sub-neg N/A

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

       2. distribute-rgt-in N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

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

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

       6. lft-mult-inverse N/A

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

       7. cancel-sign-sub-inv N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out-- N/A

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

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

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. accelerator-lowering-fma.f64 26.8

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

   13. Simplified 26.8%

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

Alternative 7: 34.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x (* y (log y))) -5e-9)
   (fma 0.5 (* z z) 0.0)
   (fma x (fma x 0.5 1.0) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + (y * log(y))) <= -5e-9) {
		tmp = fma(0.5, (z * z), 0.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(y * log(y))) <= -5e-9)
		tmp = fma(0.5, Float64(z * z), 0.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-9], N[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -5.0000000000000001e-9

   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-sub0 N/A

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

      3. --lowering--.f64 34.3

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

   5. Simplified 34.3%

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

   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 10.1

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

   8. Simplified 10.1%

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

   9. Taylor expanded in z around inf

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

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 40.7

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 40.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -5.0000000000000001e-9 < (+.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 x around inf

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

   5. Simplified 46.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 38.4

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

   8. Simplified 38.4%

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

Alternative 8: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{-1 - z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+75)
   (/ (fma (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0))) -1.0) (- -1.0 z))
   (if (<= z 4.2e+162) (exp x) (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+75) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (z * fma(z, 0.5, -1.0))), -1.0) / (-1.0 - z);
	} else if (z <= 4.2e+162) {
		tmp = exp(x);
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+75)
		tmp = Float64(fma(fma(z, 0.5, -1.0), Float64(z * Float64(z * fma(z, 0.5, -1.0))), -1.0) / Float64(-1.0 - z));
	elseif (z <= 4.2e+162)
		tmp = exp(x);
	else
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+75], N[(N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e+162], N[Exp[x], $MachinePrecision], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.59999999999999992e75

   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-sub0 N/A

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

      3. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

   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 54.2

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

   8. Simplified 54.2%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 44.0%

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

   11. Taylor expanded in z around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. --lowering--.f64 95.3

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

   13. Simplified 95.3%

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

   if -1.59999999999999992e75 < z < 4.2000000000000001e162

   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 67.2%

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

   if 4.2000000000000001e162 < 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-sub0 N/A

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

      3. --lowering--.f64 90.2

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

   5. Simplified 90.2%

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

   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 11.4

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

   8. Simplified 11.4%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 90.2%

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

Alternative 9: 73.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 4.2e+23) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+23) {
		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 <= 4.2d+23) 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 <= 4.2e+23) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.2e+23:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+23)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.2e+23)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.2e+23], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000003e23

   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 71.0%

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

   if 4.2000000000000003e23 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. log-lowering-log.f64 91.1

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

   5. Simplified 91.1%

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

   6. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 79.4

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

   8. Simplified 79.4%

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

Alternative 10: 52.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{-1 - z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= z -1.6e+75)
     (/ (fma (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0))) -1.0) (- -1.0 z))
     (if (<= z 1.5e-308)
       t_0
       (if (<= z 5.6e-170)
         (fma (* z (fma 0.25 (* z z) -1.0)) (/ (+ 2.0 (/ -4.0 z)) z) 1.0)
         (if (<= z 4.2e+162) t_0 (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	double tmp;
	if (z <= -1.6e+75) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (z * fma(z, 0.5, -1.0))), -1.0) / (-1.0 - z);
	} else if (z <= 1.5e-308) {
		tmp = t_0;
	} else if (z <= 5.6e-170) {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 + (-4.0 / z)) / z), 1.0);
	} else if (z <= 4.2e+162) {
		tmp = t_0;
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0)
	tmp = 0.0
	if (z <= -1.6e+75)
		tmp = Float64(fma(fma(z, 0.5, -1.0), Float64(z * Float64(z * fma(z, 0.5, -1.0))), -1.0) / Float64(-1.0 - z));
	elseif (z <= 1.5e-308)
		tmp = t_0;
	elseif (z <= 5.6e-170)
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 + Float64(-4.0 / z)) / z), 1.0);
	elseif (z <= 4.2e+162)
		tmp = t_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_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -1.6e+75], N[(N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e-308], t$95$0, If[LessEqual[z, 5.6e-170], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(-4.0 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 4.2e+162], t$95$0, N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.59999999999999992e75

   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-sub0 N/A

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

      3. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

   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 54.2

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

   8. Simplified 54.2%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 44.0%

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

   11. Taylor expanded in z around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. --lowering--.f64 95.3

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

   13. Simplified 95.3%

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

   if -1.59999999999999992e75 < z < 1.4999999999999999e-308 or 5.59999999999999991e-170 < z < 4.2000000000000001e162

   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 67.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 39.7

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

   8. Simplified 39.7%

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

   if 1.4999999999999999e-308 < z < 5.59999999999999991e-170

   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-sub0 N/A

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

      3. --lowering--.f64 20.2

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

   5. Simplified 20.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 20.2

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

   8. Simplified 20.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 20.2

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

   10. Applied egg-rr 20.2%

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

   11. Taylor expanded in z around inf

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

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

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

       6. distribute-neg-frac N/A

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

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

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

       8. metadata-eval 53.7

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

   13. Simplified 53.7%

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

   if 4.2000000000000001e162 < 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-sub0 N/A

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

      3. --lowering--.f64 90.2

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

   5. Simplified 90.2%

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

   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 11.4

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

   8. Simplified 11.4%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 90.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{-1 - z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot -0.16666666666666666, 0\right)}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2600.0)
   (/ 1.0 (/ 1.0 (fma z (* (* z z) -0.16666666666666666) 0.0)))
   (if (<= x 1.1e-192)
     (fma (* z (fma 0.25 (* z z) -1.0)) (/ 1.0 (fma z 0.5 1.0)) 1.0)
     (if (<= x 1.1e+93)
       (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))
       (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2600.0) {
		tmp = 1.0 / (1.0 / fma(z, ((z * z) * -0.16666666666666666), 0.0));
	} else if (x <= 1.1e-192) {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), (1.0 / fma(z, 0.5, 1.0)), 1.0);
	} else if (x <= 1.1e+93) {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2600.0)
		tmp = Float64(1.0 / Float64(1.0 / fma(z, Float64(Float64(z * z) * -0.16666666666666666), 0.0)));
	elseif (x <= 1.1e-192)
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(1.0 / fma(z, 0.5, 1.0)), 1.0);
	elseif (x <= 1.1e+93)
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2600.0], N[(1.0 / N[(1.0 / N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-192], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.1e+93], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2600

   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-sub0 N/A

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

      3. --lowering--.f64 40.5

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

   5. Simplified 40.5%

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

   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 14.4

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

   8. Simplified 14.4%

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

   9. Taylor expanded in z around inf

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

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(0 \cdot 0 - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(0 \cdot 0 - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(\color{blue}{0} - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       4. sub0-neg N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)\right)}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       5. mul0-rgt N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       6. sub-neg N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) - 0}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       7. --rgt-identity N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       8. pow2 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{2}}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{2}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + \color{blue}{0}}} \]

       10. +-rgt-identity N/A

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

       11. pow-div N/A

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

   13. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot -0.16666666666666666, 0\right)}}} \]

   if -2600 < x < 1.10000000000000003e-192

   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-sub0 N/A

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

      3. --lowering--.f64 61.8

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

   5. Simplified 61.8%

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

   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 34.5

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

   8. Simplified 34.5%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. sub-neg N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. accelerator-lowering-fma.f64 41.6

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

   10. Applied egg-rr 41.6%

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

   if 1.10000000000000003e-192 < x < 1.10000000000000011e93

   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-sub0 N/A

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

      3. --lowering--.f64 68.1

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

   5. Simplified 68.1%

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

   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 37.7

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

   8. Simplified 37.7%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 31.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 48.9%

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

   if 1.10000000000000011e93 < 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 96.0%

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

   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 88.5

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

   8. Simplified 88.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot -0.16666666666666666, 0\right)}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.5, -1\right), z \cdot \left(z \cdot \mathsf{fma}\left(z, 0.5, -1\right)\right), -1\right)}{-1 - z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+75)
   (/ (fma (fma z 0.5 -1.0) (* z (* z (fma z 0.5 -1.0))) -1.0) (- -1.0 z))
   (if (<= z 4.2e+162)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+75) {
		tmp = fma(fma(z, 0.5, -1.0), (z * (z * fma(z, 0.5, -1.0))), -1.0) / (-1.0 - z);
	} else if (z <= 4.2e+162) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+75)
		tmp = Float64(fma(fma(z, 0.5, -1.0), Float64(z * Float64(z * fma(z, 0.5, -1.0))), -1.0) / Float64(-1.0 - z));
	elseif (z <= 4.2e+162)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+75], N[(N[(N[(z * 0.5 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(z * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e+162], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.59999999999999992e75

   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-sub0 N/A

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

      3. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

   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 54.2

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

   8. Simplified 54.2%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 44.0%

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

   11. Taylor expanded in z around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. --lowering--.f64 95.3

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

   13. Simplified 95.3%

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

   if -1.59999999999999992e75 < z < 4.2000000000000001e162

   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 67.2%

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

   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 37.7

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

   8. Simplified 37.7%

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

   if 4.2000000000000001e162 < 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-sub0 N/A

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

      3. --lowering--.f64 90.2

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

   5. Simplified 90.2%

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

   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 11.4

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

   8. Simplified 11.4%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 90.2%

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

Alternative 13: 46.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot -0.16666666666666666, 0\right)}}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-189}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -550.0)
   (/ 1.0 (/ 1.0 (fma z (* (* z z) -0.16666666666666666) 0.0)))
   (if (<= x 2.3e-189)
     (fma z (* z (fma z -0.16666666666666666 0.5)) 1.0)
     (if (<= x 9e+92)
       (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))
       (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -550.0) {
		tmp = 1.0 / (1.0 / fma(z, ((z * z) * -0.16666666666666666), 0.0));
	} else if (x <= 2.3e-189) {
		tmp = fma(z, (z * fma(z, -0.16666666666666666, 0.5)), 1.0);
	} else if (x <= 9e+92) {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(1.0 / Float64(1.0 / fma(z, Float64(Float64(z * z) * -0.16666666666666666), 0.0)));
	elseif (x <= 2.3e-189)
		tmp = fma(z, Float64(z * fma(z, -0.16666666666666666, 0.5)), 1.0);
	elseif (x <= 9e+92)
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -550.0], N[(1.0 / N[(1.0 / N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-189], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 9e+92], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -550

   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-sub0 N/A

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

      3. --lowering--.f64 40.5

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

   5. Simplified 40.5%

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

   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 14.4

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

   8. Simplified 14.4%

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

   9. Taylor expanded in z around inf

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

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \color{blue}{\frac{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(0 \cdot 0 - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(0 \cdot 0 - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(\color{blue}{0} - \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       4. sub0-neg N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot 0\right)\right)}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       5. mul0-rgt N/A

          \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       6. sub-neg N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) - 0}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       7. --rgt-identity N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       8. pow2 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{2}}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + {0}^{3}}} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{2}}{{\left(\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)}^{3} + \color{blue}{0}}} \]

       10. +-rgt-identity N/A

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

       11. pow-div N/A

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

   13. Applied egg-rr 50.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(z, \left(z \cdot z\right) \cdot -0.16666666666666666, 0\right)}}} \]

   if -550 < x < 2.2999999999999998e-189

   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-sub0 N/A

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

      3. --lowering--.f64 61.8

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

   5. Simplified 61.8%

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

   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 39.3

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

   8. Simplified 39.3%

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

   9. Taylor expanded in z around inf

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. distribute-neg-in N/A

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

       4. +-commutative N/A

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

       5. sub-neg N/A

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

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. associate-*r* N/A

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

       13. sub-neg N/A

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

       14. distribute-rgt-in N/A

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

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

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

       16. associate-*l* N/A

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

   11. Simplified 39.3%

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

   if 2.2999999999999998e-189 < x < 8.9999999999999998e92

   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-sub0 N/A

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

      3. --lowering--.f64 68.1

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

   5. Simplified 68.1%

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

   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 37.7

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

   8. Simplified 37.7%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 31.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 48.9%

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

   if 8.9999999999999998e92 < 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 96.0%

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

   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 88.5

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

   8. Simplified 88.5%

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

Alternative 14: 46.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -480.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 2.8e-187)
     (fma z (* z (fma z -0.16666666666666666 0.5)) 1.0)
     (if (<= x 1.4e+88)
       (/ -1.0 (fma z (fma z 0.5 -1.0) -1.0))
       (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -480.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 2.8e-187) {
		tmp = fma(z, (z * fma(z, -0.16666666666666666, 0.5)), 1.0);
	} else if (x <= 1.4e+88) {
		tmp = -1.0 / fma(z, fma(z, 0.5, -1.0), -1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -480.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 2.8e-187)
		tmp = fma(z, Float64(z * fma(z, -0.16666666666666666, 0.5)), 1.0);
	elseif (x <= 1.4e+88)
		tmp = Float64(-1.0 / fma(z, fma(z, 0.5, -1.0), -1.0));
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -480.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-187], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.4e+88], N[(-1.0 / N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -480

   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-sub0 N/A

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

      3. --lowering--.f64 40.5

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

   5. Simplified 40.5%

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

   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 14.4

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

   8. Simplified 14.4%

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

   9. Taylor expanded in z around inf

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

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-lowering-*.f64 49.6

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

   13. Applied egg-rr 49.6%

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

   if -480 < x < 2.8e-187

   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-sub0 N/A

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

      3. --lowering--.f64 61.8

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

   5. Simplified 61.8%

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

   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 39.3

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

   8. Simplified 39.3%

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

   9. Taylor expanded in z around inf

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. distribute-neg-in N/A

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

       4. +-commutative N/A

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

       5. sub-neg N/A

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

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

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. associate-*r* N/A

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

       13. sub-neg N/A

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

       14. distribute-rgt-in N/A

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

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

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

       16. associate-*l* N/A

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

   11. Simplified 39.3%

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

   if 2.8e-187 < x < 1.39999999999999994e88

   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-sub0 N/A

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

      3. --lowering--.f64 68.1

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

   5. Simplified 68.1%

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

   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 37.7

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

   8. Simplified 37.7%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   10. Applied egg-rr 31.0%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 48.9%

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

   if 1.39999999999999994e88 < 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 96.0%

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

   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 88.5

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

   8. Simplified 88.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5500:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5500.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 3.4e+56)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5500.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 3.4e+56) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5500.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 3.4e+56)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5500.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+56], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5500

   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-sub0 N/A

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

      3. --lowering--.f64 40.5

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

   5. Simplified 40.5%

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

   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 14.4

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

   8. Simplified 14.4%

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

   9. Taylor expanded in z around inf

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

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-lowering-*.f64 49.6

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

   13. Applied egg-rr 49.6%

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

   if -5500 < x < 3.40000000000000001e56

   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-sub0 N/A

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

      3. --lowering--.f64 65.1

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

   5. Simplified 65.1%

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

   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%

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

   if 3.40000000000000001e56 < 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 94.4%

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

   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 82.2

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

   8. Simplified 82.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5500:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -420.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 9.6e+54)
     (fma z (* z (fma z -0.16666666666666666 0.5)) 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 <= -420.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 9.6e+54) {
		tmp = fma(z, (z * fma(z, -0.16666666666666666, 0.5)), 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 <= -420.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 9.6e+54)
		tmp = fma(z, Float64(z * fma(z, -0.16666666666666666, 0.5)), 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, -420.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.6e+54], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $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 < -420

   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-sub0 N/A

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

      3. --lowering--.f64 40.5

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

   5. Simplified 40.5%

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

   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 14.4

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

   8. Simplified 14.4%

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

   9. Taylor expanded in z around inf

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

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

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

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

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

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-lowering-*.f64 49.6

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

   13. Applied egg-rr 49.6%

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

   if -420 < x < 9.59999999999999993e54

   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-sub0 N/A

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

      3. --lowering--.f64 65.1

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

   5. Simplified 65.1%

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

   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

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

       1. sub-neg N/A

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

       2. remove-double-neg N/A

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

       3. distribute-neg-in N/A

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

       4. +-commutative N/A

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

       5. sub-neg N/A

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

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

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left({z}^{2} \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right), 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right), 1\right) \]

       9. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\mathsf{neg}\left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(-1 \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\left(-1 \cdot z\right) \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}, 1\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\left(-1 \cdot z\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}\right), 1\right) \]

       14. distribute-rgt-in N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right) \cdot \left(-1 \cdot z\right)\right)}, 1\right) \]

       15. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)} \cdot \left(-1 \cdot z\right)\right), 1\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{z}\right)\right) \cdot \left(-1 \cdot z\right)\right)}\right), 1\right) \]

   11. Simplified 39.5%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, 1\right) \]

   if 9.59999999999999993e54 < 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 94.4%

      \[\leadsto e^{\color{blue}{x}} \]

   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 82.2

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.16 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -480.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 7.16e+109)
     (fma z (* z (fma z -0.16666666666666666 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 <= -480.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.16e+109) {
		tmp = fma(z, (z * fma(z, -0.16666666666666666, 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 <= -480.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 7.16e+109)
		tmp = fma(z, Float64(z * fma(z, -0.16666666666666666, 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, -480.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.16e+109], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $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 < -480

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 40.5

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 40.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 14.4

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

       6. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 49.6

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 49.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot z\right)} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right) \cdot z} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right) \cdot z} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right)} \cdot z \]

       6. *-lowering-*.f64 49.6

          \[\leadsto \left(-0.16666666666666666 \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot z \]

   13. Applied egg-rr 49.6%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(z \cdot z\right)\right) \cdot z} \]

   if -480 < x < 7.1600000000000001e109

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 62.8

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 62.8%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 38.0

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{{z}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \frac{1}{6}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, {z}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right) \]

       2. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z, {z}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right) \]

       3. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z, {z}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right) + \frac{1}{6}\right)\right)\right)}, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, {z}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}\right)\right), 1\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, {z}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right), 1\right) \]

       6. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{neg}\left({z}^{2} \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right), 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right), 1\right) \]

       9. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\mathsf{neg}\left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(-1 \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}, 1\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\left(-1 \cdot z\right) \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}, 1\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\left(-1 \cdot z\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)\right)}\right), 1\right) \]

       14. distribute-rgt-in N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right) \cdot \left(-1 \cdot z\right)\right)}, 1\right) \]

       15. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(\frac{1}{z}\right)\right)\right)} \cdot \left(-1 \cdot z\right)\right), 1\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(z, z \cdot \left(\frac{1}{6} \cdot \left(-1 \cdot z\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{z}\right)\right) \cdot \left(-1 \cdot z\right)\right)}\right), 1\right) \]

   11. Simplified 37.6%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, 1\right) \]

   if 7.1600000000000001e109 < 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 95.4%

      \[\leadsto e^{\color{blue}{x}} \]

   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 80.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.16 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-26}:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.3e-26)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 7.6e+106)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.3e-26) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.6e+106) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.3e-26)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 7.6e+106)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.3e-26], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e+106], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2999999999999998e-26

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 43.7

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 43.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 17.0

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 17.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)} \cdot {z}^{3} \]

       2. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)} \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{z}^{3} \cdot \frac{1}{6}}\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) + 0} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot {z}^{3}}\right)\right) + 0 \]

       6. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {z}^{3}} + 0 \]

       7. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {z}^{3} + 0 \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {z}^{3}, 0\right)} \]

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot \left(z \cdot z\right)}, 0\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{{z}^{2}}, 0\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{z \cdot {z}^{2}}, 0\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

       13. *-lowering-*.f64 48.3

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, z \cdot \color{blue}{\left(z \cdot z\right)}, 0\right) \]

   11. Simplified 48.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, z \cdot \left(z \cdot z\right), 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot z\right)} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right) \cdot z} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right) \cdot z} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(z \cdot z\right)\right)} \cdot z \]

       6. *-lowering-*.f64 48.3

          \[\leadsto \left(-0.16666666666666666 \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot z \]

   13. Applied egg-rr 48.3%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(z \cdot z\right)\right) \cdot z} \]

   if -3.2999999999999998e-26 < x < 7.5999999999999996e106

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 61.9

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 61.9%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 35.4

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   if 7.5999999999999996e106 < 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 95.5%

      \[\leadsto e^{\color{blue}{x}} \]

   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 78.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-26}:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6e-11)
   (fma 0.5 (* z z) 0.0)
   (if (<= x 8e+106)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e-11) {
		tmp = fma(0.5, (z * z), 0.0);
	} else if (x <= 8e+106) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e-11)
		tmp = fma(0.5, Float64(z * z), 0.0);
	elseif (x <= 8e+106)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e-11], N[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[x, 8e+106], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e-11

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 42.3

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 42.3%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 14.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 39.1

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 39.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -6e-11 < x < 8.00000000000000073e106

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - z}} \]

      3. --lowering--.f64 62.0

         \[\leadsto e^{\color{blue}{0 - z}} \]

   5. Simplified 62.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   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 35.0

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 35.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   if 8.00000000000000073e106 < 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 95.5%

      \[\leadsto e^{\color{blue}{x}} \]

   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 78.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 14.1% 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 56.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 15.6

      \[\leadsto \color{blue}{1 + x} \]

8. Simplified 15.6%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 15.6%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 21: 13.9% 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 56.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 15.4%

   \[\leadsto \color{blue}{1} \]
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 2024195 
(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)))