Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.2

Time: 9.1s

Precision: binary64

Cost: 7232

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

↓

\[-6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

↓

(FPCore (x)
 :precision binary64
 (* -6.0 (/ (+ x -1.0) (- (- -1.0 x) (sqrt (* x 16.0))))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

↓

double code(double x) {
	return -6.0 * ((x + -1.0) / ((-1.0 - x) - sqrt((x * 16.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-6.0d0) * ((x + (-1.0d0)) / (((-1.0d0) - x) - sqrt((x * 16.0d0))))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

↓

public static double code(double x) {
	return -6.0 * ((x + -1.0) / ((-1.0 - x) - Math.sqrt((x * 16.0))));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

↓

def code(x):
	return -6.0 * ((x + -1.0) / ((-1.0 - x) - math.sqrt((x * 16.0))))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

↓

function code(x)
	return Float64(-6.0 * Float64(Float64(x + -1.0) / Float64(Float64(-1.0 - x) - sqrt(Float64(x * 16.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

↓

function tmp = code(x)
	tmp = -6.0 * ((x + -1.0) / ((-1.0 - x) - sqrt((x * 16.0))));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(-6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(N[(-1.0 - x), $MachinePrecision] - N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.0
Herbie	0.2

\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \]

Derivation

1. Initial program 0.2

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

2. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{-6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]
   Proof

   [Start] 0.3	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   *-commutative [=>] 0.3	\[ \color{blue}{\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(\left(x + -1\right) \cdot -6\right)} \]
   associate-*r* [=>] 0.2	\[ \color{blue}{\left(\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(x + -1\right)\right) \cdot -6} \]
   *-commutative [=>] 0.2	\[ \color{blue}{-6 \cdot \left(\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(x + -1\right)\right)} \]
   associate-*l/ [=>] 0.2	\[ -6 \cdot \color{blue}{\frac{1 \cdot \left(x + -1\right)}{-1 - \left(x + \sqrt{x \cdot 16}\right)}} \]
   *-lft-identity [=>] 0.2	\[ -6 \cdot \frac{\color{blue}{x + -1}}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   associate--r+ [=>] 0.2	\[ -6 \cdot \frac{x + -1}{\color{blue}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]

4. Final simplification 0.2

   \[\leadsto -6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}} \]

Alternatives

Alternative 1
Error	2.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.16:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\\ \end{array} \]

Alternative 2
Error	2.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 19:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6 + \sqrt{x} \cdot -24}{x}\\ \end{array} \]

Alternative 3
Error	2.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 4
Error	2.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \]

Alternative 5
Error	2.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 6
Error	32.0
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))