Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.19% → 0.14%

Time: 15.6s

Precision: binary64

Cost: 19904

Math

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

↓

\[\left(x + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(y\right)\right) + \log y \cdot \left(-0.5 - y\right)\right)\right) - z \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ x (+ (expm1 (log1p y)) (* (log y) (- -0.5 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 + (expm1(log1p(y)) + (log(y) * (-0.5 - y)))) - 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.expm1(Math.log1p(y)) + (Math.log(y) * (-0.5 - y)))) - 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.expm1(math.log1p(y)) + (math.log(y) * (-0.5 - 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(Float64(x + Float64(expm1(log1p(y)) + Float64(log(y) * Float64(-0.5 - 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[(N[(x + N[(N[(Exp[N[Log[1 + y], $MachinePrecision]] - 1), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Target

Original	0.19%
Target	0.19%
Herbie	0.14%

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

Derivation

1. Initial program 0.19

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

2. Simplified 0.19

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

   [Start] 0.19	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   associate-+l- [=>] 0.19	\[ \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z \]

3. Applied egg-rr 0.14

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

4. Simplified 0.14

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

   [Start] 0.14	\[ \left(x - \left(\left(\left(y + 0.5\right) \cdot \log y - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)\right) - z \]
   associate-+l- [=>] 0.14	\[ \left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log y - \left(e^{\mathsf{log1p}\left(y\right)} - 1\right)\right)}\right) - z \]
   *-commutative [=>] 0.14	\[ \left(x - \left(\color{blue}{\log y \cdot \left(y + 0.5\right)} - \left(e^{\mathsf{log1p}\left(y\right)} - 1\right)\right)\right) - z \]
   +-commutative [<=] 0.14	\[ \left(x - \left(\log y \cdot \color{blue}{\left(0.5 + y\right)} - \left(e^{\mathsf{log1p}\left(y\right)} - 1\right)\right)\right) - z \]

5. Applied egg-rr 0.14

   \[\leadsto \left(x - \left(\log y \cdot \left(0.5 + y\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y\right)\right)}\right)\right) - z \]

6. Final simplification 0.14

   \[\leadsto \left(x + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(y\right)\right) + \log y \cdot \left(-0.5 - y\right)\right)\right) - z \]

Alternatives

Alternative 1
Error	0.14%
Cost	13376

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

Alternative 2
Error	24.34%
Cost	7376

\[\begin{array}{l} t_0 := x - \left(y \cdot \log y - y\right)\\ \mathbf{if}\;z \leq -3.15 \cdot 10^{+140}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3
Error	24.62%
Cost	7376

\[\begin{array}{l} t_0 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 6.3 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-153}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 80000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \log y - y\right)\\ \end{array} \]

Alternative 4
Error	24.24%
Cost	7244

\[\begin{array}{l} t_0 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-153}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \log y - y\right)\\ \end{array} \]

Alternative 5
Error	10.73%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq 4700000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+55}:\\ \;\;\;\;\left(y + \log y \cdot \left(-0.5 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 6
Error	30.27%
Cost	7116

\[\begin{array}{l} t_0 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-153}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 7
Error	10.68%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq 92000000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \log y - y\right)\\ \end{array} \]

Alternative 8
Error	10.68%
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;y \leq 56000000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\left(y - t_0\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - \left(t_0 - y\right)\\ \end{array} \]

Alternative 9
Error	10.65%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq 58000000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+55}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 10
Error	0.19%
Cost	7104

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

Alternative 11
Error	28.32%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+69}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 12
Error	53.31%
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 13
Error	41.62%
Cost	192

\[x - z \]

Alternative 14
Error	69.77%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))