Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Accuracy: 99.8% → 99.9%

Time: 13.6s

Precision: binary64

Cost: 13376

Math

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

↓

\[x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

↓

(FPCore (x y z) :precision binary64 (+ x (- (fma (log y) (- -0.5 y) y) z)))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

↓

double code(double x, double y, double z) {
	return x + (fma(log(y), (-0.5 - y), y) - z);
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

↓

function code(x, y, z)
	return Float64(x + Float64(fma(log(y), Float64(-0.5 - y), y) - z))
end

Wolfram

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

↓

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

Target

Original	99.8%
Target	99.8%
Herbie	99.9%

\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

2. Simplified 99.9%

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

   [Start] 99.8	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   sub-neg [=>] 99.8	\[ \color{blue}{\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) + \left(-z\right)} \]
   sub-neg [=>] 99.8	\[ \left(\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y\right) + \left(-z\right) \]
   associate-+l+ [=>] 99.8	\[ \color{blue}{\left(x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)\right)} + \left(-z\right) \]
   associate-+l+ [=>] 99.8	\[ \color{blue}{x + \left(\left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right) + \left(-z\right)\right)} \]
   sub-neg [<=] 99.8	\[ x + \color{blue}{\left(\left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right) - z\right)} \]
   neg-sub0 [=>] 99.8	\[ x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]
   associate-+l- [=>] 99.8	\[ x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]
   neg-sub0 [<=] 99.8	\[ x + \left(\color{blue}{\left(-\left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]
   neg-mul-1 [=>] 99.8	\[ x + \left(\color{blue}{-1 \cdot \left(\left(y + 0.5\right) \cdot \log y - y\right)} - z\right) \]

3. Final simplification 99.9%

   \[\leadsto x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right) \]

Alternatives

Alternative 1
Accuracy	74.9%
Cost	7641

\[\begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 5.3 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-170}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+89}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+137} \lor \neg \left(y \leq 8.5 \cdot 10^{+187}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 2
Accuracy	74.8%
Cost	7376

\[\begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 4.4 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+57}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 3
Accuracy	89.5%
Cost	7245

\[\begin{array}{l} \mathbf{if}\;y \leq 2.46 \cdot 10^{+89}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+134} \lor \neg \left(y \leq 4.8 \cdot 10^{+188}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 4
Accuracy	89.5%
Cost	7244

\[\begin{array}{l} t_0 := y - y \cdot \log y\\ \mathbf{if}\;y \leq 2.25 \cdot 10^{+89}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+135}:\\ \;\;\;\;t_0 - z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+187}:\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 0.09:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 6
Accuracy	99.8%
Cost	7104

\[\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \]

Alternative 7
Accuracy	71.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -100 \lor \neg \left(x \leq 0.00265\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]

Alternative 8
Accuracy	48.5%
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	59.2%
Cost	192

\[x - z \]

Alternative 10
Accuracy	30.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))