Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 16.4s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ (+ (* y (- 1.0 (log y))) x) (* (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 (((y * (1.0 - log(y))) + x) + (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 = (((y * (1.0d0 - log(y))) + x) + (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 (((y * (1.0 - Math.log(y))) + x) + (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 (((y * (1.0 - math.log(y))) + x) + (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(Float64(y * Float64(1.0 - log(y))) + x) + 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 = (((y * (1.0 - log(y))) + x) + (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[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * -0.5), $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. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   associate-+l- [=>] 0.1	\[ \color{blue}{\left(x - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z \]

3. Taylor expanded in y around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	14.3
Cost	7376

\[\begin{array}{l} t_0 := x + \left(y - y \cdot \log y\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+24}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-202}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3
Error	14.8
Cost	7376

\[\begin{array}{l} t_0 := x + \left(y - y \cdot \log y\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+24}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-227}:\\ \;\;\;\;y + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4
Error	7.0
Cost	7245

\[\begin{array}{l} \mathbf{if}\;y \leq 5000000 \lor \neg \left(y \leq 3.6 \cdot 10^{+48}\right) \land y \leq 1.45 \cdot 10^{+69}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+21} \lor \neg \left(x \leq 6.2 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\left(y + x\right) - y \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(-0.5 - y\right)\right) - z\\ \end{array} \]

Alternative 6
Error	6.5
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+24}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+25}:\\ \;\;\;\;\left(y + x\right) + \log y \cdot \left(-0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 7
Error	27.4
Cost	7120

\[\begin{array}{l} t_0 := \log y \cdot -0.5\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{-306}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 8
Error	18.2
Cost	7117

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+90} \lor \neg \left(y \leq 2.9 \cdot 10^{+120}\right) \land y \leq 7 \cdot 10^{+136}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9
Error	0.1
Cost	7104

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

Alternative 10
Error	6.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 31000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \]

Alternative 11
Error	32.8
Cost	392

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 12
Error	26.6
Cost	192

\[x - z \]

Alternative 13
Error	44.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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))