Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.8s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (- (+ x (* y (- 1.0 (log y)))) (* (log y) 0.5)) 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 + (y * (1.0 - log(y)))) - (log(y) * 0.5)) - 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 = ((x + (y * (1.0d0 - log(y)))) - (log(y) * 0.5d0)) - 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 ((x + (y * (1.0 - Math.log(y)))) - (Math.log(y) * 0.5)) - z;
}

Python

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

↓

def code(x, y, z):
	return ((x + (y * (1.0 - math.log(y)))) - (math.log(y) * 0.5)) - 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(x + Float64(y * Float64(1.0 - log(y)))) - Float64(log(y) * 0.5)) - 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 = ((x + (y * (1.0 - log(y)))) - (log(y) * 0.5)) - 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[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Target

Original	99.8%
Target	99.8%
Herbie	99.9%

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 99.9%

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13632

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

Alternative 2
Accuracy	99.9%
Cost	13376

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

Alternative 3
Accuracy	74.4%
Cost	7376

\[\begin{array}{l} t_0 := \left(y + x\right) - y \cdot \log y\\ t_1 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-76}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	71.3%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-249}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-242}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+113}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \end{array} \]

Alternative 5
Accuracy	76.0%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-250}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-242}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 6
Accuracy	89.1%
Cost	7244

\[\begin{array}{l} t_0 := \left(x - \log y \cdot 0.5\right) - z\\ \mathbf{if}\;y \leq 1.46 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+72}:\\ \;\;\;\;\left(y + x\right) - y \cdot \log y\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 7
Accuracy	99.3%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 0.122:\\ \;\;\;\;\left(y + \left(x - \log y \cdot 0.5\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]

Alternative 8
Accuracy	99.8%
Cost	7104

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

Alternative 9
Accuracy	70.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -200000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 430:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 10
Accuracy	58.1%
Cost	192

\[x - z \]

Alternative 11
Accuracy	30.7%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023165 
(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))