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

Average Accuracy: 90.5% → 97.1%

Time: 24.2s

Precision: binary64

Cost: 34632

Math

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

↓

\[\begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)\\ t_1 := \log x \cdot \left(x + -0.5\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(0.91893853320467 + t_1\right) - e^{\mathsf{log1p}\left(x\right)}\right) + 1\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(t_1 - x\right)\right) + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

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 (* z (+ (* (+ y 0.0007936500793651) z) -0.0027777777777778)))
        (t_1 (* (log x) (+ x -0.5))))
   (if (<= t_0 -5e+80)
     (+ (- (* x (log x)) x) (* (+ y 0.0007936500793651) (/ (* z z) x)))
     (if (<= t_0 2e+260)
       (+
        (+ (- (+ 0.91893853320467 t_1) (exp (log1p x))) 1.0)
        (/
         (fma
          z
          (fma (+ y 0.0007936500793651) z -0.0027777777777778)
          0.083333333333333)
         x))
       (+
        (+ 0.91893853320467 (- t_1 x))
        (* 0.0007936500793651 (* z (/ z x))))))))

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 = z * (((y + 0.0007936500793651) * z) + -0.0027777777777778);
	double t_1 = log(x) * (x + -0.5);
	double tmp;
	if (t_0 <= -5e+80) {
		tmp = ((x * log(x)) - x) + ((y + 0.0007936500793651) * ((z * z) / x));
	} else if (t_0 <= 2e+260) {
		tmp = (((0.91893853320467 + t_1) - exp(log1p(x))) + 1.0) + (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + (t_1 - x)) + (0.0007936500793651 * (z * (z / x)));
	}
	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(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) + -0.0027777777777778))
	t_1 = Float64(log(x) * Float64(x + -0.5))
	tmp = 0.0
	if (t_0 <= -5e+80)
		tmp = Float64(Float64(Float64(x * log(x)) - x) + Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x)));
	elseif (t_0 <= 2e+260)
		tmp = Float64(Float64(Float64(Float64(0.91893853320467 + t_1) - exp(log1p(x))) + 1.0) + Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(t_1 - x)) + Float64(0.0007936500793651 * Float64(z * Float64(z / x))));
	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[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] + -0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+80], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+260], N[(N[(N[(N[(0.91893853320467 + t$95$1), $MachinePrecision] - N[Exp[N[Log[1 + x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(t$95$1 - x), $MachinePrecision]), $MachinePrecision] + N[(0.0007936500793651 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	90.5%
Target	98.1%
Herbie	97.1%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < -4.99999999999999961e80

   1. Initial program 70.1%

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

   2. Taylor expanded in z around inf 70.1%

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

   3. Simplified 89.4%

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

      [Start] 70.1	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x} \]
      associate-/l* [=>] 85.5	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}} \]
      associate-/r/ [=>] 89.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{x} \cdot \left(0.0007936500793651 + y\right)} \]
      unpow2 [=>] 89.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{z \cdot z}}{x} \cdot \left(0.0007936500793651 + y\right) \]

   4. Taylor expanded in x around inf 89.5%

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

   5. Simplified 89.4%

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

      [Start] 89.5	\[ \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      *-commutative [=>] 89.5	\[ \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      sub-neg [=>] 89.5	\[ x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      metadata-eval [=>] 89.5	\[ x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      distribute-lft-in [=>] 89.4	\[ \color{blue}{\left(x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + x \cdot -1\right)} + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      mul-1-neg [=>] 89.4	\[ \left(x \cdot \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + x \cdot -1\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      log-rec [=>] 89.4	\[ \left(x \cdot \left(-\color{blue}{\left(-\log x\right)}\right) + x \cdot -1\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      remove-double-neg [=>] 89.4	\[ \left(x \cdot \color{blue}{\log x} + x \cdot -1\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      *-commutative [=>] 89.4	\[ \left(\color{blue}{\log x \cdot x} + x \cdot -1\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      *-commutative [<=] 89.4	\[ \left(\log x \cdot x + \color{blue}{-1 \cdot x}\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      neg-mul-1 [<=] 89.4	\[ \left(\log x \cdot x + \color{blue}{\left(-x\right)}\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      unsub-neg [=>] 89.4	\[ \color{blue}{\left(\log x \cdot x - x\right)} + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]
      *-commutative [<=] 89.4	\[ \left(\color{blue}{x \cdot \log x} - x\right) + \frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right) \]

   if -4.99999999999999961e80 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < 2.00000000000000013e260

   1. Initial program 99.4%

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

   2. Simplified 99.5%

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

      [Start] 99.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      associate-+l- [=>] 99.4	\[ \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - \left(x - 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      sub-neg [=>] 99.4	\[ \left(\left(x - 0.5\right) \cdot \log x - \color{blue}{\left(x + \left(-0.91893853320467\right)\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      associate--r+ [=>] 99.4	\[ \color{blue}{\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) - \left(-0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      associate--r+ [<=] 99.4	\[ \color{blue}{\left(\left(x - 0.5\right) \cdot \log x - \left(x + \left(-0.91893853320467\right)\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      sub-neg [<=] 99.4	\[ \left(\left(x - 0.5\right) \cdot \log x - \color{blue}{\left(x - 0.91893853320467\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      fma-neg [=>] 99.5	\[ \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, -\left(x - 0.91893853320467\right)\right)} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      sub-neg [=>] 99.5	\[ \mathsf{fma}\left(\color{blue}{x + \left(-0.5\right)}, \log x, -\left(x - 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      metadata-eval [=>] 99.5	\[ \mathsf{fma}\left(x + \color{blue}{-0.5}, \log x, -\left(x - 0.91893853320467\right)\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      neg-sub0 [=>] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0 - \left(x - 0.91893853320467\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      associate-+l- [<=] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right) + 0.91893853320467}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      neg-sub0 [<=] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(-x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      +-commutative [=>] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0.91893853320467 + \left(-x\right)}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
      unsub-neg [=>] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0.91893853320467 - x}\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - e^{\mathsf{log1p}\left(x\right)}\right) + 1\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      Proof

      [Start] 99.5	\[ \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      fma-udef [=>] 99.4	\[ \color{blue}{\left(\left(x + -0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      associate-+r- [=>] 99.4	\[ \color{blue}{\left(\left(\left(x + -0.5\right) \cdot \log x + 0.91893853320467\right) - x\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      expm1-log1p-u [=>] 99.5	\[ \left(\left(\left(x + -0.5\right) \cdot \log x + 0.91893853320467\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      expm1-udef [=>] 99.5	\[ \left(\left(\left(x + -0.5\right) \cdot \log x + 0.91893853320467\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)}\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      associate--r- [=>] 99.5	\[ \color{blue}{\left(\left(\left(\left(x + -0.5\right) \cdot \log x + 0.91893853320467\right) - e^{\mathsf{log1p}\left(x\right)}\right) + 1\right)} + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
      +-commutative [=>] 99.5	\[ \left(\left(\color{blue}{\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right)} - e^{\mathsf{log1p}\left(x\right)}\right) + 1\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   if 2.00000000000000013e260 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)

   1. Initial program 25.3%

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

   2. Taylor expanded in z around inf 23.7%

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

   3. Simplified 38.4%

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

      [Start] 23.7	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x} \]
      associate-/l* [=>] 38.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}} \]
      associate-/r/ [=>] 38.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{x} \cdot \left(0.0007936500793651 + y\right)} \]
      unpow2 [=>] 38.4	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{z \cdot z}}{x} \cdot \left(0.0007936500793651 + y\right) \]

   4. Taylor expanded in y around 0 25.7%

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

   5. Simplified 81.3%

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

      [Start] 25.7	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.0007936500793651 \cdot \frac{{z}^{2}}{x} \]
      unpow2 [=>] 25.7	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.0007936500793651 \cdot \frac{\color{blue}{z \cdot z}}{x} \]
      associate-*r/ [<=] 81.3	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + 0.0007936500793651 \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right) \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(\left(\left(0.91893853320467 + \log x \cdot \left(x + -0.5\right)\right) - e^{\mathsf{log1p}\left(x\right)}\right) + 1\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.1%
Cost	27976

\[\begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x} + \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.1%
Cost	9160

\[\begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)\\ t_1 := 0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_1 + \frac{t_0 + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1 + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 3
Accuracy	95.6%
Cost	9032

\[\begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;\left(x \cdot \log x - x\right) + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{t_0 + 0.083333333333333}{x} + x \cdot \left(-1 - \log \left(\frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 4
Accuracy	95.5%
Cost	8904

\[\begin{array}{l} t_0 := z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z + -0.0027777777777778\right)\\ t_1 := x \cdot \log x - x\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;t_1 + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_1 + \frac{t_0 + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 5
Accuracy	91.5%
Cost	7756

\[\begin{array}{l} t_0 := x \cdot \log x - x\\ t_1 := t_0 + z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-22}:\\ \;\;\;\;t_0 + \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \leq 10:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{\frac{1}{x}}{12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	91.5%
Cost	7756

\[\begin{array}{l} t_0 := x \cdot \log x - x\\ t_1 := t_0 + z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-21}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x} + \left(t_0 + 0.91893853320467\right)\\ \mathbf{elif}\;z \leq 10.6:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{\frac{1}{x}}{12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	91.4%
Cost	7756

\[\begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\\ t_1 := x \cdot \log x - x\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + 0.0007936500793651 \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x} + \left(t_1 + 0.91893853320467\right)\\ \mathbf{elif}\;z \leq 10.5:\\ \;\;\;\;t_0 + \frac{\frac{1}{x}}{12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t_1 + z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	89.1%
Cost	7624

\[\begin{array}{l} t_0 := x \cdot \log x - x\\ t_1 := z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -0.00095:\\ \;\;\;\;\left(t_0 + 0.91893853320467\right) + t_1\\ \mathbf{elif}\;z \leq 11:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_1\\ \end{array} \]

Alternative 9
Accuracy	89.1%
Cost	7624

\[\begin{array}{l} t_0 := x \cdot \log x - x\\ t_1 := z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -0.00095:\\ \;\;\;\;\left(t_0 + 0.91893853320467\right) + t_1\\ \mathbf{elif}\;z \leq 10.5:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{\frac{1}{x}}{12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_1\\ \end{array} \]

Alternative 10
Accuracy	89.1%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00095 \lor \neg \left(z \leq 12\right):\\ \;\;\;\;\left(x \cdot \log x - x\right) + z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 11
Accuracy	89.1%
Cost	7496

\[\begin{array}{l} t_0 := x \cdot \log x - x\\ t_1 := z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \mathbf{if}\;z \leq -0.00095:\\ \;\;\;\;\left(t_0 + 0.91893853320467\right) + t_1\\ \mathbf{elif}\;z \leq 10.5:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x + -0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_1\\ \end{array} \]

Alternative 12
Accuracy	80.5%
Cost	7232

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

Alternative 13
Accuracy	79.1%
Cost	7104

\[\left(\left(x \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

Alternative 14
Accuracy	79.1%
Cost	7104

\[x \cdot \left(-1 - \log \left(\frac{1}{x}\right)\right) + \frac{0.083333333333333}{x} \]

Alternative 15
Accuracy	79.1%
Cost	6976

\[\left(x \cdot \log x - x\right) + \frac{0.083333333333333}{x} \]

Alternative 16
Accuracy	33.1%
Cost	6656

\[{\left(x \cdot 12.000000000000048\right)}^{-1} \]

Alternative 17
Accuracy	33.0%
Cost	320

\[0.083333333333333 \cdot \frac{1}{x} \]

Alternative 18
Accuracy	33.0%
Cost	192

\[\frac{0.083333333333333}{x} \]

Reproduce

herbie shell --seed 2023138 
(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)))