Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 17.1s

Precision: binary64

Cost: 20032

Math

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

↓

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

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)) (+ (* (log y) (- -0.5 y)) -1.0))) 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)) + ((log(y) * (-0.5 - y)) + -1.0))) - 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)) + ((Math.log(y) * (-0.5 - y)) + -1.0))) - 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)) + ((math.log(y) * (-0.5 - y)) + -1.0))) - 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(Float64(log(y) * Float64(-0.5 - y)) + -1.0))) - 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[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision]), $MachinePrecision] + -1.0), $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 0.1

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

4. Simplified 0.1

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

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	14.5
Cost	7508

\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) - z\\ t_1 := x - y \cdot \left(\log y + -1\right)\\ \mathbf{if}\;y \leq 1780000:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+91}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.4
Cost	7245

\[\begin{array}{l} \mathbf{if}\;y \leq 3700000:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+38} \lor \neg \left(y \leq 2.6 \cdot 10^{+72}\right):\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4
Error	14.4
Cost	7245

\[\begin{array}{l} \mathbf{if}\;y \leq 1780000:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+38} \lor \neg \left(y \leq 1.15 \cdot 10^{+73}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5
Error	7.3
Cost	7244

\[\begin{array}{l} t_0 := \left(x + \log y \cdot -0.5\right) - z\\ \mathbf{if}\;y \leq 3.2 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 6
Error	7.2
Cost	7244

\[\begin{array}{l} t_0 := \left(x + \log y \cdot -0.5\right) - z\\ \mathbf{if}\;y \leq 9 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;\left(y + \log y \cdot \left(-0.5 - y\right)\right) - z\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 7
Error	0.1
Cost	7104

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

Alternative 8
Error	18.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+123}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9
Error	32.7
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{+87}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	26.4
Cost	192

\[x - z \]

Alternative 11
Error	44.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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))