Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.0

Time: 11.1s

Precision: binary64

Cost: 7232

Math

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

↓

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

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 (* 4.0 (sqrt x)))))))

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 + (4.0 * sqrt(x)))));
}

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 + (4.0d0 * sqrt(x)))))
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 + (4.0 * Math.sqrt(x)))));
}

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 + (4.0 * math.sqrt(x)))))

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(-1.0 - Float64(x + Float64(4.0 * sqrt(x))))))
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 + (4.0 * sqrt(x)))));
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[(-1.0 - N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Simplified 0.0

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

   [Start] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   rational.json-simplify-6 [<=] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   rational.json-simplify-2 [=>] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right) \cdot 1}} \]
   metadata-eval [<=] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-1 \cdot -1\right)}} \]
   rational.json-simplify-43 [<=] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\color{blue}{-1 \cdot \left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right) \cdot -1\right)}} \]
   rational.json-simplify-2 [<=] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{-1 \cdot \color{blue}{\left(-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)}} \]
   rational.json-simplify-47 [<=] 0.2	\[ \color{blue}{\frac{\frac{6 \cdot \left(x - 1\right)}{-1}}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   rational.json-simplify-49 [=>] 0.2	\[ \frac{\color{blue}{\left(x - 1\right) \cdot \frac{6}{-1}}}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\left(x - 1\right) \cdot \color{blue}{-6}}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   metadata-eval [<=] 0.2	\[ \frac{\left(x - 1\right) \cdot \color{blue}{\left(-6\right)}}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   rational.json-simplify-49 [=>] 0.0	\[ \color{blue}{\left(-6\right) \cdot \frac{x - 1}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   metadata-eval [=>] 0.0	\[ \color{blue}{-6} \cdot \frac{x - 1}{-1 \cdot \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   rational.json-simplify-47 [<=] 0.0	\[ -6 \cdot \color{blue}{\frac{\frac{x - 1}{-1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]
   rational.json-simplify-10 [<=] 0.0	\[ -6 \cdot \frac{\color{blue}{-\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   rational.json-simplify-1 [=>] 0.0	\[ -6 \cdot \frac{-\left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]
   rational.json-simplify-1 [=>] 0.0	\[ -6 \cdot \frac{-\left(x - 1\right)}{4 \cdot \sqrt{x} + \color{blue}{\left(1 + x\right)}} \]
   rational.json-simplify-41 [=>] 0.0	\[ -6 \cdot \frac{-\left(x - 1\right)}{\color{blue}{1 + \left(x + 4 \cdot \sqrt{x}\right)}} \]
   rational.json-simplify-17 [=>] 0.0	\[ -6 \cdot \frac{-\left(x - 1\right)}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right) - -1}} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.6
Cost	7236

\[\begin{array}{l} t_0 := -1 - \left(x + 4 \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{6}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -6}{t_0}\\ \end{array} \]

Alternative 2
Error	2.9
Cost	1344

\[-6 \cdot \left(2 \cdot \frac{\frac{-0.5}{1 - x}}{\frac{\frac{-1 - x}{x + -1}}{x + -1}}\right) \]

Alternative 3
Error	2.9
Cost	1088

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

Alternative 4
Error	2.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 0.49:\\ \;\;\;\;-6 \cdot \left(1 + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{6}{x} - 6\right)\\ \end{array} \]

Alternative 5
Error	2.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;6 \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4}{x} + -2\right) \cdot -3\\ \end{array} \]

Alternative 6
Error	2.9
Cost	576

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

Alternative 7
Error	2.9
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq 0.49:\\ \;\;\;\;12 \cdot x - 6\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{6}{x} - 6\right)\\ \end{array} \]

Alternative 8
Error	2.9
Cost	452

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

Alternative 9
Error	2.9
Cost	196

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

Alternative 10
Error	33.5
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023075 
(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)))))