Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Accuracy: 99.8% → 99.9%

Time: 16.9s

Precision: binary64

Cost: 20032

Math

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

↓

\[\left(x + \left(e^{\mathsf{log1p}\left(y\right)} + \left(-1 - \log y \cdot \left(y + 0.5\right)\right)\right)\right) - z \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ x (+ (exp (log1p y)) (- -1.0 (* (log y) (+ y 0.5))))) 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 + (exp(log1p(y)) + (-1.0 - (log(y) * (y + 0.5))))) - z;
}

Java

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

↓

public static double code(double x, double y, double z) {
	return (x + (Math.exp(Math.log1p(y)) + (-1.0 - (Math.log(y) * (y + 0.5))))) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

↓

def code(x, y, z):
	return (x + (math.exp(math.log1p(y)) + (-1.0 - (math.log(y) * (y + 0.5))))) - 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(Float64(x + Float64(exp(log1p(y)) + Float64(-1.0 - Float64(log(y) * Float64(y + 0.5))))) - 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[(N[(x + N[(N[Exp[N[Log[1 + y], $MachinePrecision]], $MachinePrecision] + N[(-1.0 - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $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 99.8%

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

   [Start] 99.8	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   associate-+l- [=>] 99.8	\[ \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z \]

3. Applied egg-rr 99.9%

   \[\leadsto \left(x - \color{blue}{\left(\left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)}\right) - z \]
   Proof

   [Start] 99.8	\[ \left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right) - z \]
   expm1-log1p-u [=>] 99.9	\[ \left(x - \left(\left(y + 0.5\right) \cdot \log y - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y\right)\right)}\right)\right) - z \]
   expm1-udef [=>] 99.9	\[ \left(x - \left(\left(y + 0.5\right) \cdot \log y - \color{blue}{\left(e^{\mathsf{log1p}\left(y\right)} - 1\right)}\right)\right) - z \]
   associate--r- [=>] 99.9	\[ \left(x - \color{blue}{\left(\left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)}\right) - z \]

4. Simplified 99.9%

   \[\leadsto \left(x - \color{blue}{\left(\left(1 + \log y \cdot \left(0.5 + y\right)\right) - e^{\mathsf{log1p}\left(y\right)}\right)}\right) - z \]
   Proof

   [Start] 99.9	\[ \left(x - \left(\left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)\right) - z \]
   +-commutative [=>] 99.9	\[ \left(x - \color{blue}{\left(1 + \left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right)\right)}\right) - z \]
   associate-+r- [=>] 99.9	\[ \left(x - \color{blue}{\left(\left(1 + \left(y + 0.5\right) \cdot \log y\right) - e^{\mathsf{log1p}\left(y\right)}\right)}\right) - z \]
   *-commutative [=>] 99.9	\[ \left(x - \left(\left(1 + \color{blue}{\log y \cdot \left(y + 0.5\right)}\right) - e^{\mathsf{log1p}\left(y\right)}\right)\right) - z \]
   +-commutative [<=] 99.9	\[ \left(x - \left(\left(1 + \log y \cdot \color{blue}{\left(0.5 + y\right)}\right) - e^{\mathsf{log1p}\left(y\right)}\right)\right) - z \]

5. Final simplification 99.9%

   \[\leadsto \left(x + \left(e^{\mathsf{log1p}\left(y\right)} + \left(-1 - \log y \cdot \left(y + 0.5\right)\right)\right)\right) - z \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13376

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

Alternative 2
Accuracy	76.0%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-69}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-27}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 3
Accuracy	71.0%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-69}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+108}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+103} \lor \neg \left(z \leq 9 \cdot 10^{+45}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	7104

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

Alternative 6
Accuracy	99.8%
Cost	7104

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

Alternative 7
Accuracy	89.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 8
Accuracy	71.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+106}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9
Accuracy	46.7%
Cost	656

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+24}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+159}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	57.9%
Cost	192

\[x - z \]

Alternative 11
Accuracy	30.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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))