Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 12.5s

Precision: binary64

Cost: 7616

Math

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

↓

\[\left(\frac{y}{-2} + \left(y - \left(\log y \cdot \left(0.5 + y\right) + \left(\frac{y}{-2} - x\right)\right)\right)\right) - z \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ (/ y -2.0) (- y (+ (* (log y) (+ 0.5 y)) (- (/ y -2.0) x)))) 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 / -2.0) + (y - ((log(y) * (0.5 + y)) + ((y / -2.0) - x)))) - 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 / (-2.0d0)) + (y - ((log(y) * (0.5d0 + y)) + ((y / (-2.0d0)) - x)))) - 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 / -2.0) + (y - ((Math.log(y) * (0.5 + y)) + ((y / -2.0) - x)))) - z;
}

Python

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

↓

def code(x, y, z):
	return ((y / -2.0) + (y - ((math.log(y) * (0.5 + y)) + ((y / -2.0) - x)))) - 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 / -2.0) + Float64(y - Float64(Float64(log(y) * Float64(0.5 + y)) + Float64(Float64(y / -2.0) - x)))) - 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 / -2.0) + (y - ((log(y) * (0.5 + y)) + ((y / -2.0) - x)))) - 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 / -2.0), $MachinePrecision] + N[(y - N[(N[(N[Log[y], $MachinePrecision] * N[(0.5 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / -2.0), $MachinePrecision] - x), $MachinePrecision]), $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. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\left(\frac{y}{2} - \left(\left(\left(y + 0.5\right) \cdot \log y - x\right) - \frac{y}{2}\right)\right)} - z \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(\left(y - \left(\left(y + 0.5\right) \cdot \log y + \left(\frac{y}{-2} - x\right)\right)\right) + \frac{y}{-2}\right)} - z \]

4. Simplified 0.1

   \[\leadsto \color{blue}{\left(\frac{y}{-2} + \left(y - \left(\log y \cdot \left(0.5 + y\right) + \left(\frac{y}{-2} - x\right)\right)\right)\right)} - z \]
   Proof

   [Start] 0.1	\[ \left(\left(y - \left(\left(y + 0.5\right) \cdot \log y + \left(\frac{y}{-2} - x\right)\right)\right) + \frac{y}{-2}\right) - z \]
   rational_best-simplify-3 [=>] 0.1	\[ \color{blue}{\left(\frac{y}{-2} + \left(y - \left(\left(y + 0.5\right) \cdot \log y + \left(\frac{y}{-2} - x\right)\right)\right)\right)} - z \]
   rational_best-simplify-3 [<=] 0.1	\[ \left(\frac{y}{-2} + \left(y - \color{blue}{\left(\left(\frac{y}{-2} - x\right) + \left(y + 0.5\right) \cdot \log y\right)}\right)\right) - z \]
   rational_best-simplify-59 [=>] 0.1	\[ \left(\frac{y}{-2} + \left(y - \color{blue}{\left(\left(y + 0.5\right) \cdot \log y - \left(-\left(\frac{y}{-2} - x\right)\right)\right)}\right)\right) - z \]
   rational_best-simplify-59 [<=] 0.1	\[ \left(\frac{y}{-2} + \left(y - \color{blue}{\left(\left(\frac{y}{-2} - x\right) + \left(y + 0.5\right) \cdot \log y\right)}\right)\right) - z \]
   rational_best-simplify-3 [=>] 0.1	\[ \left(\frac{y}{-2} + \left(y - \color{blue}{\left(\left(y + 0.5\right) \cdot \log y + \left(\frac{y}{-2} - x\right)\right)}\right)\right) - z \]
   rational_best-simplify-1 [=>] 0.1	\[ \left(\frac{y}{-2} + \left(y - \left(\color{blue}{\log y \cdot \left(y + 0.5\right)} + \left(\frac{y}{-2} - x\right)\right)\right)\right) - z \]
   rational_best-simplify-3 [=>] 0.1	\[ \left(\frac{y}{-2} + \left(y - \left(\log y \cdot \color{blue}{\left(0.5 + y\right)} + \left(\frac{y}{-2} - x\right)\right)\right)\right) - z \]

5. Final simplification 0.1

   \[\leadsto \left(\frac{y}{-2} + \left(y - \left(\log y \cdot \left(0.5 + y\right) + \left(\frac{y}{-2} - x\right)\right)\right)\right) - z \]

Alternatives

Alternative 1
Error	18.0
Cost	7508

\[\begin{array}{l} t_0 := y - \log y \cdot \left(0.5 + y\right)\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-215}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-183}:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(y + y\right) - \left(-x\right)\right) - z\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{y}{-2} + x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7360

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

Alternative 3
Error	10.0
Cost	7244

\[\begin{array}{l} t_0 := y - \log y \cdot \left(0.5 + y\right)\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{+31}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{y}{-2} + x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.065:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;\left(y - \left(0.5 + y\right) \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{-2} + x\right) - z\\ \end{array} \]

Alternative 5
Error	0.1
Cost	7104

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

Alternative 6
Error	6.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+21}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(-\log y\right)\right) - z\\ \end{array} \]

Alternative 7
Error	18.4
Cost	6984

\[\begin{array}{l} t_0 := \left(\frac{y}{-2} + x\right) - z\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.036:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	25.2
Cost	6856

\[\begin{array}{l} t_0 := \left(\frac{y}{-2} + x\right) - z\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-85}:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	24.7
Cost	448

\[\left(\frac{y}{-2} + x\right) - z \]

Alternative 10
Error	26.6
Cost	192

\[x - z \]

Alternative 11
Error	44.6
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023100 
(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))