Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.18% → 0.13%

Time: 13.1s

Precision: binary64

Cost: 13376

Math

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

↓

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

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 + fma(log(y), Float64(-0.5 - y), Float64(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[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.18%
Target	0.18%
Herbie	0.13%

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

Derivation

1. Initial program 0.18

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

2. Simplified 0.13

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

   [Start] 0.18	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   associate--l+ [=>] 0.18	\[ \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]
   cancel-sign-sub-inv [=>] 0.18	\[ \color{blue}{\left(x + \left(-\left(y + 0.5\right)\right) \cdot \log y\right)} + \left(y - z\right) \]
   associate-+l+ [=>] 0.18	\[ \color{blue}{x + \left(\left(-\left(y + 0.5\right)\right) \cdot \log y + \left(y - z\right)\right)} \]
   *-commutative [=>] 0.18	\[ x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]
   fma-def [=>] 0.13	\[ x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]
   neg-sub0 [=>] 0.13	\[ x + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y - z\right) \]
   +-commutative [=>] 0.13	\[ x + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y - z\right) \]
   associate--r+ [=>] 0.13	\[ x + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y - z\right) \]
   metadata-eval [=>] 0.13	\[ x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

3. Final simplification 0.13

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

Alternatives

Alternative 1
Error	10.93%
Cost	7504

\[\begin{array}{l} t_0 := 1 - \log y\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+78}:\\ \;\;\;\;x + y \cdot t_0\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{t_0}{\frac{1}{y}}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+70}:\\ \;\;\;\;\left(y + \log y \cdot \left(-0.5 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 2
Error	28.49%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+78} \lor \neg \left(y \leq 1.35 \cdot 10^{+89}\right) \land y \leq 10^{+148}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3
Error	23.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-21} \lor \neg \left(z \leq 1.62 \cdot 10^{+75}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 4
Error	10.28%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.18%
Cost	7104

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

Alternative 6
Error	10.27%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+52}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 7
Error	10.25%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+51}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 8
Error	40.06%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;y \leq 4.05 \cdot 10^{+176}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\log y\right)\\ \end{array} \]

Alternative 9
Error	51.26%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	41.6%
Cost	192

\[x - z \]

Alternative 11
Error	69.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(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))