Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 9.6s

Precision: binary64

Cost: 13632

Math

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

↓

\[\left(\left(\left(1 - \log y\right) \cdot y + x\right) - 0.5 \cdot \log y\right) - z \]

FPCore

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

↓

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((1.0d0 - log(y)) * y) + x) - (0.5d0 * log(y))) - z
end function

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 ((((1.0 - Math.log(y)) * y) + x) - (0.5 * Math.log(y))) - z;
}

Python

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

↓

def code(x, y, z):
	return ((((1.0 - math.log(y)) * y) + x) - (0.5 * math.log(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(Float64(Float64(Float64(1.0 - log(y)) * y) + x) - Float64(0.5 * log(y))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

↓

function tmp = code(x, y, z)
	tmp = ((((1.0 - log(y)) * y) + x) - (0.5 * log(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[(N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - N[(0.5 * N[Log[y], $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. Taylor expanded in y around 0 0.1

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

3. Final simplification 0.1

   \[\leadsto \left(\left(\left(1 - \log y\right) \cdot y + x\right) - 0.5 \cdot \log y\right) - z \]

Alternatives

Alternative 1
Error	19.1
Cost	7380

\[\begin{array}{l} t_0 := y - \left(0.5 + y\right) \cdot \log y\\ \mathbf{if}\;x \leq -960:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -3.65 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-261}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 290:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7360

\[\left(\left(y + \left(x + \left(1 - \left(y + 0.5\right) \cdot \log y\right)\right)\right) - 1\right) - z \]

Alternative 3
Error	18.5
Cost	7248

\[\begin{array}{l} t_0 := -0.5 \cdot \log y - z\\ \mathbf{if}\;x \leq -14000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-261}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-278}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;x \leq 245:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4
Error	6.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+71}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;\left(y - z\right) - \log y \cdot \left(0.5 + y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5
Error	6.7
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+78}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\log \left(\frac{1}{y}\right) - -1\right) - z\\ \end{array} \]

Alternative 6
Error	0.1
Cost	7104

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

Alternative 7
Error	9.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+146}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 8
Error	26.5
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-156}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-223}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 9
Error	17.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+146}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 10
Error	26.0
Cost	192

\[x - z \]

Alternative 11
Error	44.5
Cost	128

\[-z \]

Alternative 12
Error	62.4
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023074 
(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))