Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.2

Time: 9.1s

Precision: binary64

Cost: 7616

Math

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

↓

\[\frac{1 - x}{\frac{1 + \frac{x}{\frac{\sqrt{x \cdot 16} - x}{16 - x}}}{-6}} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (- 1.0 x) (/ (+ 1.0 (/ x (/ (- (sqrt (* x 16.0)) x) (- 16.0 x)))) -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 (1.0 - x) / ((1.0 + (x / ((sqrt((x * 16.0)) - x) / (16.0 - x)))) / -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 = (1.0d0 - x) / ((1.0d0 + (x / ((sqrt((x * 16.0d0)) - x) / (16.0d0 - x)))) / (-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 (1.0 - x) / ((1.0 + (x / ((Math.sqrt((x * 16.0)) - x) / (16.0 - x)))) / -6.0);
}

Python

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

↓

def code(x):
	return (1.0 - x) / ((1.0 + (x / ((math.sqrt((x * 16.0)) - x) / (16.0 - x)))) / -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(1.0 - x) / Float64(Float64(1.0 + Float64(x / Float64(Float64(sqrt(Float64(x * 16.0)) - x) / Float64(16.0 - x)))) / -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 = (1.0 - x) / ((1.0 + (x / ((sqrt((x * 16.0)) - x) / (16.0 - x)))) / -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[(1.0 - x), $MachinePrecision] / N[(N[(1.0 + N[(x / N[(N[(N[Sqrt[N[(x * 16.0), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision] / N[(16.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -6.0), $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. Applied egg-rr 0.0

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

5. Applied egg-rr 15.8

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

6. Simplified 0.2

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

   [Start] 15.8	\[ \frac{1 - x}{\frac{1 + \frac{x \cdot 16 - x \cdot x}{\sqrt{x \cdot 16} - x}}{-6}} \]
   distribute-lft-out-- [=>] 15.8	\[ \frac{1 - x}{\frac{1 + \frac{\color{blue}{x \cdot \left(16 - x\right)}}{\sqrt{x \cdot 16} - x}}{-6}} \]
   associate-/l* [=>] 0.2	\[ \frac{1 - x}{\frac{1 + \color{blue}{\frac{x}{\frac{\sqrt{x \cdot 16} - x}{16 - x}}}}{-6}} \]

7. Final simplification 0.2

   \[\leadsto \frac{1 - x}{\frac{1 + \frac{x}{\frac{\sqrt{x \cdot 16} - x}{16 - x}}}{-6}} \]

Alternatives

Alternative 1
Error	2.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\frac{1 - x}{\frac{1 + x}{-6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{4 \cdot \sqrt{x} + \left(1 + x\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7232

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

Alternative 3
Error	0.0
Cost	7232

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

Alternative 4
Error	0.0
Cost	7232

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

Alternative 5
Error	2.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1 - x}{-0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{x + -1}}\\ \end{array} \]

Alternative 6
Error	2.9
Cost	580

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

Alternative 7
Error	2.9
Cost	576

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

Alternative 8
Error	2.9
Cost	452

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

Alternative 9
Error	2.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1 - x}{-0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-6}{x}\\ \end{array} \]

Alternative 10
Error	2.9
Cost	196

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

Alternative 11
Error	33.3
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2022364 
(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)))))