Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Accuracy: 90.7% → 97.0%

Time: 33.2s

Precision: binary64

Cost: 15048

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

↓

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{+116}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{\frac{1}{x}}{12.000000000000048} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (+ x -0.5) (log x)) x) 0.91893853320467)))
   (if (<= z -9.4e+116)
     (* z (* z (/ (+ y 0.0007936500793651) x)))
     (if (<= z 1.32e+154)
       (+
        t_0
        (+
         (/ y (/ x (* z z)))
         (+
          (/ (/ 1.0 x) 12.000000000000048)
          (/ (fma 0.0007936500793651 z -0.0027777777777778) (/ x z)))))
       (+ t_0 (* (/ z x) (* z (+ y 0.0007936500793651))))))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

↓

double code(double x, double y, double z) {
	double t_0 = (((x + -0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (z <= -9.4e+116) {
		tmp = z * (z * ((y + 0.0007936500793651) / x));
	} else if (z <= 1.32e+154) {
		tmp = t_0 + ((y / (x / (z * z))) + (((1.0 / x) / 12.000000000000048) + (fma(0.0007936500793651, z, -0.0027777777777778) / (x / z))));
	} else {
		tmp = t_0 + ((z / x) * (z * (y + 0.0007936500793651)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (z <= -9.4e+116)
		tmp = Float64(z * Float64(z * Float64(Float64(y + 0.0007936500793651) / x)));
	elseif (z <= 1.32e+154)
		tmp = Float64(t_0 + Float64(Float64(y / Float64(x / Float64(z * z))) + Float64(Float64(Float64(1.0 / x) / 12.000000000000048) + Float64(fma(0.0007936500793651, z, -0.0027777777777778) / Float64(x / z)))));
	else
		tmp = Float64(t_0 + Float64(Float64(z / x) * Float64(z * Float64(y + 0.0007936500793651))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[z, -9.4e+116], N[(z * N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.32e+154], N[(t$95$0 + N[(N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / x), $MachinePrecision] / 12.000000000000048), $MachinePrecision] + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(z / x), $MachinePrecision] * N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	90.7%
Target	98.0%
Herbie	97.0%

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \]

Derivation

1. Split input into 3 regimes

2. if z < -9.4000000000000007e116

   1. Initial program 30.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Simplified 30.5%

      \[\leadsto \color{blue}{0.91893853320467 + \left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} - \mathsf{fma}\left(\log x, 0.5 - x, x\right)\right)} \]
      Proof

      [Start] 30.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      +-commutative [=>] 30.4	\[ \color{blue}{\left(0.91893853320467 + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      associate-+l+ [=>] 30.4	\[ \color{blue}{0.91893853320467 + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\right)} \]
      +-commutative [<=] 30.4	\[ 0.91893853320467 + \color{blue}{\left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(x - 0.5\right) \cdot \log x - x\right)\right)} \]
      sub-neg [=>] 30.4	\[ 0.91893853320467 + \left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \color{blue}{\left(\left(x - 0.5\right) \cdot \log x + \left(-x\right)\right)}\right) \]
      +-commutative [=>] 30.4	\[ 0.91893853320467 + \left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \color{blue}{\left(\left(-x\right) + \left(x - 0.5\right) \cdot \log x\right)}\right) \]
      associate-+r+ [=>] 30.4	\[ 0.91893853320467 + \color{blue}{\left(\left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(-x\right)\right) + \left(x - 0.5\right) \cdot \log x\right)} \]
      unsub-neg [=>] 30.4	\[ 0.91893853320467 + \left(\color{blue}{\left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} - x\right)} + \left(x - 0.5\right) \cdot \log x\right) \]
      associate-+l- [=>] 30.4	\[ 0.91893853320467 + \color{blue}{\left(\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} - \left(x - \left(x - 0.5\right) \cdot \log x\right)\right)} \]
      remove-double-neg [<=] 30.4	\[ 0.91893853320467 + \left(\color{blue}{\left(-\left(-\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\right)\right)} - \left(x - \left(x - 0.5\right) \cdot \log x\right)\right) \]
      neg-mul-1 [=>] 30.4	\[ 0.91893853320467 + \left(\color{blue}{-1 \cdot \left(-\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\right)} - \left(x - \left(x - 0.5\right) \cdot \log x\right)\right) \]
      *-commutative [<=] 30.4	\[ 0.91893853320467 + \left(\color{blue}{\left(-\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\right) \cdot -1} - \left(x - \left(x - 0.5\right) \cdot \log x\right)\right) \]

   3. Taylor expanded in z around inf 24.0%

      \[\leadsto \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]

   4. Simplified 55.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)} \]
      Proof

      [Start] 24.0	\[ \left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2} \]
      *-commutative [=>] 24.0	\[ \color{blue}{{z}^{2} \cdot \left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right)} \]
      unpow2 [=>] 24.0	\[ \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \]
      associate-*l* [=>] 55.5	\[ \color{blue}{z \cdot \left(z \cdot \left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right)\right)} \]
      associate-*r/ [=>] 55.5	\[ z \cdot \left(z \cdot \left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651 \cdot 1}{x}}\right)\right) \]
      metadata-eval [=>] 55.5	\[ z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{\color{blue}{0.0007936500793651}}{x}\right)\right) \]

   5. Taylor expanded in x around 0 55.5%

      \[\leadsto z \cdot \left(z \cdot \color{blue}{\frac{0.0007936500793651 + y}{x}}\right) \]

   6. Simplified 55.5%

      \[\leadsto z \cdot \left(z \cdot \color{blue}{\frac{y + 0.0007936500793651}{x}}\right) \]
      Proof

      [Start] 55.5	\[ z \cdot \left(z \cdot \frac{0.0007936500793651 + y}{x}\right) \]
      +-commutative [<=] 55.5	\[ z \cdot \left(z \cdot \frac{\color{blue}{y + 0.0007936500793651}}{x}\right) \]

   if -9.4000000000000007e116 < z < 1.31999999999999998e154

   1. Initial program 96.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in y around 0 96.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y \cdot {z}^{2}}{x} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right)} \]

   3. Simplified 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right)} \]
      Proof

      [Start] 96.8	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y \cdot {z}^{2}}{x} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
      associate-/l* [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\color{blue}{\frac{y}{\frac{x}{{z}^{2}}}} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
      unpow2 [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{\color{blue}{z \cdot z}}} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
      associate-*r/ [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{x}} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
      metadata-eval [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{\color{blue}{0.083333333333333}}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
      associate-/l* [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \color{blue}{\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{\frac{x}{z}}}\right)\right) \]
      fma-neg [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\color{blue}{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}}{\frac{x}{z}}\right)\right) \]
      metadata-eval [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\mathsf{fma}\left(0.0007936500793651, z, \color{blue}{-0.0027777777777778}\right)}{\frac{x}{z}}\right)\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{{\left(x \cdot 12.000000000000048\right)}^{-1}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      Proof

      [Start] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      clear-num [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{1}{\frac{x}{0.083333333333333}}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      inv-pow [=>] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      div-inv [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left({\color{blue}{\left(x \cdot \frac{1}{0.083333333333333}\right)}}^{-1} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      metadata-eval [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left({\left(x \cdot \color{blue}{12.000000000000048}\right)}^{-1} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{\frac{1}{x}}{12.000000000000048}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      Proof

      [Start] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left({\left(x \cdot 12.000000000000048\right)}^{-1} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      unpow-1 [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{1}{x \cdot 12.000000000000048}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]
      associate-/r* [=>] 99.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{\frac{1}{x}}{12.000000000000048}} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right) \]

   if 1.31999999999999998e154 < z

   1. Initial program 2.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in z around inf 0.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]

   3. Simplified 94.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{z}{x} \cdot \left(z \cdot \left(0.0007936500793651 + y\right)\right)} \]
      Proof

      [Start] 0.0	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x} \]
      +-commutative [<=] 0.0	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{{z}^{2} \cdot \color{blue}{\left(y + 0.0007936500793651\right)}}{x} \]
      associate-/l* [=>] 0.0	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{\frac{x}{y + 0.0007936500793651}}} \]
      unpow2 [=>] 0.0	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{z \cdot z}}{\frac{x}{y + 0.0007936500793651}} \]
      associate-/l* [=>] 99.1	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{z}{\frac{\frac{x}{y + 0.0007936500793651}}{z}}} \]
      associate-/r* [<=] 94.8	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z}{\color{blue}{\frac{x}{\left(y + 0.0007936500793651\right) \cdot z}}} \]
      associate-/r/ [=>] 94.8	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{z}{x} \cdot \left(\left(y + 0.0007936500793651\right) \cdot z\right)} \]
      *-commutative [=>] 94.8	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z}{x} \cdot \color{blue}{\left(z \cdot \left(y + 0.0007936500793651\right)\right)} \]
      +-commutative [=>] 94.8	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z}{x} \cdot \left(z \cdot \color{blue}{\left(0.0007936500793651 + y\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.4 \cdot 10^{+116}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{\frac{1}{x}}{12.000000000000048} + \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.2%
Cost	23752

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := z \cdot \left(y + 0.0007936500793651\right)\\ t_2 := t_0 + \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + t_1\right)}{x}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;t_0 + y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;t_2 \leq 10^{+302}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{z}{x} \cdot t_1\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	23752

\[\begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x\\ t_1 := \left(t_0 - x\right) + 0.91893853320467\\ t_2 := z \cdot \left(y + 0.0007936500793651\right)\\ t_3 := \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + t_2\right)}{x}\\ t_4 := t_1 + t_3\\ \mathbf{if}\;t_4 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;t_1 + y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;t_4 \leq 10^{+302}:\\ \;\;\;\;t_3 + \left(\left(0.91893853320467 - x\right) + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{z}{x} \cdot t_2\\ \end{array} \]

Alternative 3
Accuracy	98.2%
Cost	23752

\[\begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x\\ t_1 := \left(t_0 - x\right) + 0.91893853320467\\ t_2 := z \cdot \left(y + 0.0007936500793651\right)\\ t_3 := \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + t_2\right)}{x}\\ t_4 := t_1 + t_3\\ \mathbf{if}\;t_4 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;t_1 + y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;t_4 \leq 10^{+302}:\\ \;\;\;\;\left(\left(0.91893853320467 + t_3\right) - x\right) + t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{z}{x} \cdot t_2\\ \end{array} \]

Alternative 4
Accuracy	97.0%
Cost	14920

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{+116}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{\frac{x}{z}} + \frac{0.083333333333333}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	8904

\[\begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := z \cdot \left(-0.0027777777777778 + t_0\right)\\ t_2 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+194}:\\ \;\;\;\;t_2 + y \cdot \frac{z}{\frac{x}{z}}\\ \mathbf{elif}\;t_1 \leq 6000000000000:\\ \;\;\;\;t_2 + \frac{0.083333333333333 + z \cdot \left(z \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{z}{x} \cdot t_0\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	8388

\[\begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ t_1 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;z \cdot \left(-0.0027777777777778 + t_0\right) \leq 6000000000000:\\ \;\;\;\;t_1 + \left(\frac{y}{\frac{x}{z \cdot z}} + \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{z}{x} \cdot t_0\\ \end{array} \]

Alternative 7
Accuracy	94.1%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;x \leq 1650:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \frac{z}{\frac{x}{z}}\\ \end{array} \]

Alternative 8
Accuracy	97.3%
Cost	7748

\[\begin{array}{l} t_0 := z \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + t_0\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z}{x} \cdot t_0\\ \end{array} \]

Alternative 9
Accuracy	85.5%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+116} \lor \neg \left(z \leq 3.1 \cdot 10^{+102}\right):\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot y\right)}{x} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 10
Accuracy	93.7%
Cost	7620

\[\begin{array}{l} \mathbf{if}\;x \leq 2050:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + y \cdot \frac{z}{\frac{x}{z}}\\ \end{array} \]

Alternative 11
Accuracy	87.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+108}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]

Alternative 12
Accuracy	87.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\left(\left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right) + \left(x + -0.5\right) \cdot \log x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+108}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]

Alternative 13
Accuracy	87.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\left(\frac{0.083333333333333}{x} + \left(x + -0.5\right) \cdot \log x\right) + \left(0.91893853320467 - x\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+108}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x\\ \end{array} \]

Alternative 14
Accuracy	85.2%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+88} \lor \neg \left(x \leq 3.8 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \log x - x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\left(y + 0.0007936500793651\right) \cdot \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	50.7%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+69}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{elif}\;z \leq 88000000:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{1}{\frac{-0.0007936500793651 - y}{\frac{-1}{\frac{1}{x}}}}}\\ \end{array} \]

Alternative 16
Accuracy	50.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+69} \lor \neg \left(z \leq 5.4 \cdot 10^{+38}\right):\\ \;\;\;\;z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ \end{array} \]

Alternative 17
Accuracy	19.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 0.055\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 18
Accuracy	19.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+14} \lor \neg \left(y \leq 0.1\right):\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 19
Accuracy	19.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 20
Accuracy	19.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 21
Accuracy	19.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 22
Accuracy	18.9%
Cost	576

\[z \cdot \left(z \cdot \frac{y + 0.0007936500793651}{x}\right) \]

Alternative 23
Accuracy	19.0%
Cost	576

\[z \cdot \frac{z}{\frac{x}{y + 0.0007936500793651}} \]

Alternative 24
Accuracy	20.3%
Cost	576

\[\left(y + 0.0007936500793651\right) \cdot \frac{z}{\frac{x}{z}} \]

Alternative 25
Accuracy	11.4%
Cost	448

\[0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right) \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))