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

Average Error: 9.96% → 0.72%

Time: 17.7s

Precision: binary64

Cost: 14656

Math

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

↓

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

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

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

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)
	return Float64(Float64(Float64(Float64(Float64(x + -0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(z * Float64(z / x)) * y) + Float64(Float64(0.083333333333333 / x) + Float64(fma(0.0007936500793651, z, -0.0027777777777778) / Float64(x / z)))))
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_] := N[(N[(N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	9.96%
Target	2.07%
Herbie	0.72%

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

Derivation

1. Initial program 9.96

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

2. Taylor expanded in y around 0 10.09

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

3. Simplified 7.02

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

   [Start] 10.09	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y \cdot {z}^{2}}{x} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
   associate-/l* [=>] 7.04	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\color{blue}{\frac{y}{\frac{x}{{z}^{2}}}} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
   unpow2 [=>] 7.04	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{\color{blue}{z \cdot z}}} + \left(0.083333333333333 \cdot \frac{1}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
   associate-*r/ [=>] 7.02	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\color{blue}{\frac{0.083333333333333 \cdot 1}{x}} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
   metadata-eval [=>] 7.02	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{\color{blue}{0.083333333333333}}{x} + \frac{\left(0.0007936500793651 \cdot z - 0.0027777777777778\right) \cdot z}{x}\right)\right) \]
   associate-/l* [=>] 7.02	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \color{blue}{\frac{0.0007936500793651 \cdot z - 0.0027777777777778}{\frac{x}{z}}}\right)\right) \]
   fma-neg [=>] 7.02	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\color{blue}{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}}{\frac{x}{z}}\right)\right) \]
   metadata-eval [=>] 7.02	\[ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(\frac{y}{\frac{x}{z \cdot z}} + \left(\frac{0.083333333333333}{x} + \frac{\mathsf{fma}\left(0.0007936500793651, z, \color{blue}{-0.0027777777777778}\right)}{\frac{x}{z}}\right)\right) \]

4. Applied egg-rr 0.72

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

5. Final simplification 0.72

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

Alternatives

Alternative 1
Error	5.08%
Cost	24072

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := \left(z \cdot \frac{z}{x}\right) \cdot y\\ t_2 := \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ t_3 := t_0 + t_2\\ \mathbf{if}\;t_3 \leq 10^{+17}:\\ \;\;\;\;t_0 + \left(t_1 + \left(\frac{0.083333333333333}{x} + \frac{z \cdot -0.0027777777777778}{x}\right)\right)\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t_2 + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(\frac{0.083333333333333}{x} + \frac{0.0027777777777778 + z \cdot -0.0007936500793651}{\frac{x}{-z}}\right)\right) + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \end{array} \]

Alternative 2
Error	3.04%
Cost	24008

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ t_1 := \frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x}\\ t_2 := t_0 + t_1\\ t_3 := \frac{0.083333333333333}{x} + \frac{z \cdot -0.0027777777777778}{x}\\ \mathbf{if}\;t_2 \leq 10^{+17}:\\ \;\;\;\;t_0 + \left(\left(z \cdot \frac{z}{x}\right) \cdot y + t_3\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_1 + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(t_3 + z \cdot \left(z \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	3.77%
Cost	9417

\[\begin{array}{l} t_0 := z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\\ t_1 := \left(x + -0.5\right) \cdot \log x\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;\left(\left(t_1 - x\right) + 0.91893853320467\right) + \left(\left(\frac{0.083333333333333}{x} + \frac{z \cdot -0.0027777777777778}{x}\right) + z \cdot \left(z \cdot \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + t_0}{x} + \left(\left(t_1 + 0.91893853320467\right) - x\right)\\ \end{array} \]

Alternative 4
Error	5.28%
Cost	9161

\[\begin{array}{l} t_0 := z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+111} \lor \neg \left(t_0 \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;\left(0.91893853320467 - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + t_0}{x}\\ \end{array} \]

Alternative 5
Error	5.28%
Cost	9161

\[\begin{array}{l} t_0 := z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+111} \lor \neg \left(t_0 \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;\left(0.91893853320467 - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + t_0}{x} + \left(\left(\left(x + -0.5\right) \cdot \log x + 0.91893853320467\right) - x\right)\\ \end{array} \]

Alternative 6
Error	7.2%
Cost	9033

\[\begin{array}{l} t_0 := z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-11} \lor \neg \left(t_0 \leq 2 \cdot 10^{+243}\right):\\ \;\;\;\;\left(0.91893853320467 - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333 + t_0}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \end{array} \]

Alternative 7
Error	0.72%
Cost	8448

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

Alternative 8
Error	1.4%
Cost	8256

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

Alternative 9
Error	5.65%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 10
Error	16.51%
Cost	7752

\[\begin{array}{l} t_0 := \left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;x \leq 8.8 \cdot 10^{+43}:\\ \;\;\;\;t_0 + \frac{1}{x \cdot 12.000000000000048}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+180}:\\ \;\;\;\;t_0 + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \end{array} \]

Alternative 11
Error	6.95%
Cost	7752

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+180}:\\ \;\;\;\;\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \end{array} \]

Alternative 12
Error	7.46%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;x \leq 28:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(-0.0027777777777778 + z \cdot \left(y + 0.0007936500793651\right)\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)\right) + \frac{z}{x} \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 13
Error	18.77%
Cost	7360

\[\left(\left(\left(x + -0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{x \cdot 12.000000000000048} \]

Alternative 14
Error	18.81%
Cost	7232

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

Alternative 15
Error	18.81%
Cost	7232

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

Alternative 16
Error	20.1%
Cost	7104

\[\frac{0.083333333333333}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right) \]

Alternative 17
Error	67.24%
Cost	6656

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

Alternative 18
Error	67.28%
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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)))