Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 9.8s

Precision: binary64

Cost: 13632

Math

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

↓

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

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(1.0 - log(y)) * y) + x) - Float64(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[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - N[(N[(0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $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. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \color{blue}{\left(y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\right)} - z \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.1	\[ \color{blue}{\left(\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\right)} - z \]
   rational_best_oopsla_all_46_json_45_simplify-105 [=>] 0.1	\[ \color{blue}{\left(\left(x + y\right) - z\right) - \left(y + 0.5\right) \cdot \log y} \]

3. Taylor expanded in y around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	32.8
Cost	7380

\[\begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-266}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-98}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;y \leq 1700000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]

Alternative 2
Error	25.7
Cost	7380

\[\begin{array}{l} t_0 := x - 0.5 \cdot \log y\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-115}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;y \leq 54000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+53}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]

Alternative 3
Error	25.2
Cost	7380

\[\begin{array}{l} t_0 := 0.5 \cdot \log y\\ t_1 := x - t_0\\ t_2 := -\left(t_0 + z\right)\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 18500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]

Alternative 4
Error	6.9
Cost	7240

\[\begin{array}{l} t_0 := \left(0.5 + y\right) \cdot \log y\\ \mathbf{if}\;y \leq 4700000000:\\ \;\;\;\;x - \left(0.5 \cdot \log y + z\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+55}:\\ \;\;\;\;y - \left(t_0 + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - t_0\\ \end{array} \]

Alternative 5
Error	6.8
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 13800000000:\\ \;\;\;\;x - \left(0.5 \cdot \log y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(\left(0.5 + y\right) \cdot \log y + z\right)\\ \end{array} \]

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	0.1
Cost	7104

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

Alternative 8
Error	34.7
Cost	7052

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-30}:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	10.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;x - \left(0.5 \cdot \log y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]

Alternative 10
Error	34.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+95}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11
Error	44.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))