Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H

Average Error: 0.25% → 0.25%

Time: 4.4s

Precision: binary64

Cost: 448

Math

\[\frac{x \cdot x - 3}{6} \]

↓

\[\frac{x \cdot x + -3}{6} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

↓

(FPCore (x) :precision binary64 (/ (+ (* x x) -3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

↓

double code(double x) {
	return ((x * x) + -3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) + (-3.0d0)) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

↓

public static double code(double x) {
	return ((x * x) + -3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

↓

def code(x):
	return ((x * x) + -3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

↓

function code(x)
	return Float64(Float64(Float64(x * x) + -3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

↓

function tmp = code(x)
	tmp = ((x * x) + -3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

↓

code[x_] := N[(N[(N[(x * x), $MachinePrecision] + -3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 0.25

   \[\frac{x \cdot x - 3}{6} \]

2. Final simplification 0.25

   \[\leadsto \frac{x \cdot x + -3}{6} \]

Alternatives

Alternative 1
Error	1.65%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{6}\\ \end{array} \]

Alternative 2
Error	1.78%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]

Alternative 3
Error	1.61%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.05:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 4
Error	0.3%
Cost	448

\[-0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666 \]

Alternative 5
Error	33.31%
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))