Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.18% → 0.14%

Time: 15.9s

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	0.14%

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

Derivation

1. Initial program 0.18

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

2. Simplified 0.14

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]
   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 [=>] 0.14	\[ x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]
   neg-sub0 [=>] 0.14	\[ x + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y - z\right) \]
   +-commutative [=>] 0.14	\[ x + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y - z\right) \]
   associate--r+ [=>] 0.14	\[ x + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y - z\right) \]
   metadata-eval [=>] 0.14	\[ x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

3. Final simplification 0.14

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

Alternatives

Alternative 1
Error	30.89%
Cost	7381

\[\begin{array}{l} t_0 := x - \left(z - y\right)\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+99} \lor \neg \left(y \leq 1.75 \cdot 10^{+109}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 2
Error	23.9%
Cost	7376

\[\begin{array}{l} t_0 := x - \left(z - y\right)\\ t_1 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+107}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \]

Alternative 3
Error	24.58%
Cost	7244

\[\begin{array}{l} t_0 := x - \left(z - y\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 4
Error	24.49%
Cost	7244

\[\begin{array}{l} t_0 := x - \left(z - y\right)\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-35}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 5
Error	29.81%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+67}:\\ \;\;\;\;x - \left(z - y\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+99} \lor \neg \left(y \leq 3.7 \cdot 10^{+108}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 6
Error	10.18%
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 1.66 \cdot 10^{+53}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \]

Alternative 7
Error	0.84%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - y \cdot \log y\right)\right) - z\\ \end{array} \]

Alternative 8
Error	0.18%
Cost	7104

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

Alternative 9
Error	51.29%
Cost	656

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 830000:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	40.87%
Cost	192

\[x - z \]

Alternative 11
Error	70.02%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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))