Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 14.0s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (- (+ (+ y x) (* (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 ((y + x) + (log(y) * (-0.5 - 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 + x) + (log(y) * ((-0.5d0) - 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 + x) + (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 ((y + x) + (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(Float64(y + x) + Float64(log(y) * Float64(-0.5 - 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 + x) + (log(y) * (-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[(N[(y + x), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - 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 x around 0 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	18.9
Cost	8036

\[\begin{array}{l} t_0 := y + \log y \cdot \left(-0.5 - y\right)\\ t_1 := \left(y + x\right) - y \cdot \log y\\ t_2 := \log y \cdot -0.5 - z\\ \mathbf{if}\;x \leq -1.337444864352237 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.207865328760678 \cdot 10^{-6}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.1983935482744094 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.266062730980963 \cdot 10^{-232}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4520980573474881 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.632652901577313 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.491256334149763 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.201876436423276 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.648951932021571 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 2
Error	18.6
Cost	7640

\[\begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ t_1 := y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{if}\;x \leq -2.207865328760678 \cdot 10^{-6}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -1.1983935482744094 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.266062730980963 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4520980573474881 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.632652901577313 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 106149.10111454684:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3
Error	15.0
Cost	7640

\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;x \leq -89200507650947790:\\ \;\;\;\;\left(y + x\right) - y \cdot \log y\\ \mathbf{elif}\;x \leq -7.577282750133622 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.1653272361369644 \cdot 10^{-113}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 1.404452259899533 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3625098350616391 \cdot 10^{-70}:\\ \;\;\;\;y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;x \leq 1.0496618541294643 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4
Error	6.5
Cost	7376

\[\begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;z \leq -9.7622658752323 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3616549050694537 \cdot 10^{+23}:\\ \;\;\;\;\left(y + x\right) + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 3.0649588738958706 \cdot 10^{+100}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;z \leq 6.572736590201557 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5
Error	18.4
Cost	7116

\[\begin{array}{l} t_0 := y - y \cdot \log y\\ \mathbf{if}\;y \leq 1.9192526666117098 \cdot 10^{+77}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 9.521314549466506 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.7343966809390062 \cdot 10^{+138}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 0.013433598345464105:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) - y \cdot \log y\right) - z\\ \end{array} \]

Alternative 7
Error	6.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7577233807787693 \cdot 10^{+63}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - y \cdot \log y\\ \end{array} \]

Alternative 8
Error	33.0
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -9.7622658752323 \cdot 10^{+48}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 17160525298.810358:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	26.6
Cost	192

\[x - z \]

Alternative 10
Error	44.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(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))