Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 22.8s

Precision: binary64

Cost: 13888

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (- (+ (* y (- 1.0 (* -1.0 (log (/ 1.0 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 (((y * (1.0 - (-1.0 * log((1.0 / 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 = (((y * (1.0d0 - ((-1.0d0) * log((1.0d0 / 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 (((y * (1.0 - (-1.0 * Math.log((1.0 / 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 (((y * (1.0 - (-1.0 * math.log((1.0 / 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(y * Float64(1.0 - Float64(-1.0 * log(Float64(1.0 / 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 = (((y * (1.0 - (-1.0 * log((1.0 / 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[(y * N[(1.0 - N[(-1.0 * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in y around inf 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13632

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

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.1
Cost	13376

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

Alternative 4
Error	8.6
Cost	7504

\[\begin{array}{l} t_0 := \left(y + x\right) - \left(0.5 + y\right) \cdot \log y\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+106}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+132}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5
Error	6.5
Cost	7240

\[\begin{array}{l} t_0 := \left(0.5 + y\right) \cdot \log y\\ t_1 := \left(y - z\right) - t_0\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+21}:\\ \;\;\;\;\left(y + x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-224}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+110}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(0.5 + y\right) \cdot \log y\\ \end{array} \]

Alternative 7
Error	0.1
Cost	7104

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

Alternative 8
Error	19.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -320000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 9
Error	10.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+110}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(0.5 + y\right) \cdot \log y\\ \end{array} \]

Alternative 10
Error	32.8
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+68}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11
Error	26.8
Cost	192

\[x - z \]

Alternative 12
Error	44.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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))