Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.0

Time: 9.8s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (x + (-1.0d0)) / ((((-1.0d0) - x) + (sqrt(x) * (-4.0d0))) / (-6.0d0))
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 (x + -1.0) / (((-1.0 - x) + (Math.sqrt(x) * -4.0)) / -6.0);
}

Python

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

↓

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

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(Float64(x + -1.0) / Float64(Float64(Float64(-1.0 - x) + Float64(sqrt(x) * -4.0)) / -6.0))
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 = (x + -1.0) / (((-1.0 - x) + (sqrt(x) * -4.0)) / -6.0);
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[(N[(x + -1.0), $MachinePrecision] / N[(N[(N[(-1.0 - x), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / -6.0), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.0
Herbie	0.0

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

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 0.1

   \[\leadsto \color{blue}{-6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]
   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/ [=>] 0.1	\[ -6 \cdot \color{blue}{\frac{1 \cdot \left(x + -1\right)}{-1 - \left(x + \sqrt{x \cdot 16}\right)}} \]
   *-lft-identity [=>] 0.1	\[ -6 \cdot \frac{\color{blue}{x + -1}}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   associate--r+ [=>] 0.1	\[ -6 \cdot \frac{x + -1}{\color{blue}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.5
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 0.076:\\ \;\;\;\;-6 + \left(x \cdot -6 + \sqrt{x} \cdot 24\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\left(x + 1\right) + \sqrt{x} \cdot 4}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7232

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

Alternative 3
Error	2.1
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 + \left(x \cdot -6 + \sqrt{x} \cdot 24\right)\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 4
Error	2.7
Cost	2240

\[\begin{array}{l} t_0 := \frac{x}{x + -1}\\ \frac{6}{t_0 \cdot t_0 - \frac{\frac{1}{x + -1}}{x + -1}} \cdot \left(t_0 + \frac{-1}{x + -1}\right) \end{array} \]

Alternative 5
Error	2.7
Cost	576

\[\frac{6}{\frac{x + 1}{x + -1}} \]

Alternative 6
Error	2.7
Cost	452

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

Alternative 7
Error	2.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 8
Error	2.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6 + x \cdot 12\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 9
Error	2.7
Cost	196

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

Alternative 10
Error	32.8
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023016 
(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)))))