Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 13.4s

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	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   associate-+l- [=>] 0.1	\[ \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z \]

3. Applied egg-rr 21.6

   \[\leadsto \left(x - \left(\color{blue}{\frac{\left(0.125 + {y}^{3}\right) \cdot \log y}{0.25 + y \cdot \left(y + -0.5\right)}} - y\right)\right) - z \]

4. Applied egg-rr 0.1

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

5. Simplified 0.1

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

   [Start] 0.1	\[ \left(x - \left(\left(\log y \cdot \left(y + 0.5\right) - e^{\mathsf{log1p}\left(y\right)}\right) + 1\right)\right) - z \]
   associate-+l- [=>] 0.1	\[ \left(x - \color{blue}{\left(\log y \cdot \left(y + 0.5\right) - \left(e^{\mathsf{log1p}\left(y\right)} - 1\right)\right)}\right) - z \]
   +-commutative [=>] 0.1	\[ \left(x - \left(\log y \cdot \color{blue}{\left(0.5 + y\right)} - \left(e^{\mathsf{log1p}\left(y\right)} - 1\right)\right)\right) - z \]
   expm1-def [=>] 0.1	\[ \left(x - \left(\log y \cdot \left(0.5 + y\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y\right)\right)}\right)\right) - z \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	15.3
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+57}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3
Error	7.3
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq 215:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+41}:\\ \;\;\;\;\left(y + \log y \cdot \left(-0.5 - y\right)\right) - z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4
Error	19.1
Cost	7116

\[\begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-66}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+172}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 5
Error	0.1
Cost	7104

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

Alternative 6
Error	0.1
Cost	7104

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

Alternative 7
Error	6.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 8
Error	27.2
Cost	6856

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-234}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-297}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 9
Error	18.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+174}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 10
Error	25.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+208}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\log y\right)\\ \end{array} \]

Alternative 11
Error	33.0
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+90}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	26.9
Cost	192

\[x - z \]

Alternative 13
Error	44.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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))