Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Accuracy: 99.7% → 99.9%

Time: 10.6s

Precision: binary64

Cost: 7232

Math

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

↓

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

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 (* 4.0 (sqrt 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 + (4.0 * sqrt(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 + (4.0d0 * sqrt(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 + (4.0 * Math.sqrt(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 + (4.0 * math.sqrt(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(4.0 * sqrt(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 + (4.0 * sqrt(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[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -6.0), $MachinePrecision]), $MachinePrecision]

Target

Original	99.7%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 99.7%

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

2. Applied egg-rr 99.5%

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

   [Start] 99.7	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   frac-2neg [=>] 99.7	\[ \color{blue}{\frac{-6 \cdot \left(x - 1\right)}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   div-inv [=>] 99.6	\[ \color{blue}{\left(-6 \cdot \left(x - 1\right)\right) \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   *-commutative [=>] 99.6	\[ \left(-\color{blue}{\left(x - 1\right) \cdot 6}\right) \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   distribute-rgt-neg-in [=>] 99.6	\[ \color{blue}{\left(\left(x - 1\right) \cdot \left(-6\right)\right)} \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   sub-neg [=>] 99.6	\[ \left(\color{blue}{\left(x + \left(-1\right)\right)} \cdot \left(-6\right)\right) \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   metadata-eval [=>] 99.6	\[ \left(\left(x + \color{blue}{-1}\right) \cdot \left(-6\right)\right) \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   metadata-eval [=>] 99.6	\[ \left(\left(x + -1\right) \cdot \color{blue}{-6}\right) \cdot \frac{1}{-\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   neg-sub0 [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\color{blue}{0 - \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]
   metadata-eval [<=] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\color{blue}{\log 1} - \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)} \]
   +-commutative [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)}} \]
   associate-+r+ [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(\left(4 \cdot \sqrt{x} + x\right) + 1\right)}} \]
   +-commutative [<=] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(1 + \left(4 \cdot \sqrt{x} + x\right)\right)}} \]
   associate--r+ [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\color{blue}{\left(\log 1 - 1\right) - \left(4 \cdot \sqrt{x} + x\right)}} \]
   metadata-eval [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\left(\color{blue}{0} - 1\right) - \left(4 \cdot \sqrt{x} + x\right)} \]
   metadata-eval [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{\color{blue}{-1} - \left(4 \cdot \sqrt{x} + x\right)} \]
   +-commutative [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \color{blue}{\left(x + 4 \cdot \sqrt{x}\right)}} \]
   add-sqr-sqrt [=>] 99.6	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \color{blue}{\sqrt{4 \cdot \sqrt{x}} \cdot \sqrt{4 \cdot \sqrt{x}}}\right)} \]
   sqrt-unprod [=>] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \color{blue}{\sqrt{\left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)}}\right)} \]
   *-commutative [=>] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{\color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)}\right)} \]
   *-commutative [=>] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{\left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}}\right)} \]
   swap-sqr [=>] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}}\right)} \]
   add-sqr-sqrt [<=] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{\color{blue}{x} \cdot \left(4 \cdot 4\right)}\right)} \]
   metadata-eval [=>] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{x \cdot \color{blue}{16}}\right)} \]

3. Simplified 99.7%

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

   [Start] 99.5	\[ \left(\left(x + -1\right) \cdot -6\right) \cdot \frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   *-commutative [=>] 99.5	\[ \color{blue}{\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(\left(x + -1\right) \cdot -6\right)} \]
   associate-*r* [=>] 99.6	\[ \color{blue}{\left(\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(x + -1\right)\right) \cdot -6} \]
   *-commutative [=>] 99.6	\[ \color{blue}{-6 \cdot \left(\frac{1}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \cdot \left(x + -1\right)\right)} \]
   associate-*l/ [=>] 99.7	\[ -6 \cdot \color{blue}{\frac{1 \cdot \left(x + -1\right)}{-1 - \left(x + \sqrt{x \cdot 16}\right)}} \]
   *-lft-identity [=>] 99.7	\[ -6 \cdot \frac{\color{blue}{x + -1}}{-1 - \left(x + \sqrt{x \cdot 16}\right)} \]
   associate--r+ [=>] 99.7	\[ -6 \cdot \frac{x + -1}{\color{blue}{\left(-1 - x\right) - \sqrt{x \cdot 16}}} \]

4. Applied egg-rr 99.9%

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

   [Start] 99.7	\[ -6 \cdot \frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}} \]
   *-commutative [=>] 99.7	\[ \color{blue}{\frac{x + -1}{\left(-1 - x\right) - \sqrt{x \cdot 16}} \cdot -6} \]
   frac-2neg [=>] 99.7	\[ \color{blue}{\frac{-\left(x + -1\right)}{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}} \cdot -6 \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{\left(-\left(x + -1\right)\right) \cdot -6}{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}} \]
   associate-/l* [=>] 99.7	\[ \color{blue}{\frac{-\left(x + -1\right)}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}}} \]
   neg-sub0 [=>] 99.7	\[ \frac{\color{blue}{0 - \left(x + -1\right)}}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   metadata-eval [<=] 99.7	\[ \frac{\color{blue}{\log 1} - \left(x + -1\right)}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   +-commutative [=>] 99.7	\[ \frac{\log 1 - \color{blue}{\left(-1 + x\right)}}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   associate--r+ [=>] 99.7	\[ \frac{\color{blue}{\left(\log 1 - -1\right) - x}}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   metadata-eval [=>] 99.7	\[ \frac{\left(\color{blue}{0} - -1\right) - x}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   metadata-eval [=>] 99.7	\[ \frac{\color{blue}{1} - x}{\frac{-\left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   neg-sub0 [=>] 99.7	\[ \frac{1 - x}{\frac{\color{blue}{0 - \left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}}{-6}} \]
   metadata-eval [<=] 99.7	\[ \frac{1 - x}{\frac{\color{blue}{\log 1} - \left(\left(-1 - x\right) - \sqrt{x \cdot 16}\right)}{-6}} \]
   associate--l- [=>] 99.7	\[ \frac{1 - x}{\frac{\log 1 - \color{blue}{\left(-1 - \left(x + \sqrt{x \cdot 16}\right)\right)}}{-6}} \]
   associate--r- [=>] 99.7	\[ \frac{1 - x}{\frac{\color{blue}{\left(\log 1 - -1\right) + \left(x + \sqrt{x \cdot 16}\right)}}{-6}} \]
   metadata-eval [=>] 99.7	\[ \frac{1 - x}{\frac{\left(\color{blue}{0} - -1\right) + \left(x + \sqrt{x \cdot 16}\right)}{-6}} \]
   metadata-eval [=>] 99.7	\[ \frac{1 - x}{\frac{\color{blue}{1} + \left(x + \sqrt{x \cdot 16}\right)}{-6}} \]
   sqrt-prod [=>] 99.9	\[ \frac{1 - x}{\frac{1 + \left(x + \color{blue}{\sqrt{x} \cdot \sqrt{16}}\right)}{-6}} \]
   *-commutative [=>] 99.9	\[ \frac{1 - x}{\frac{1 + \left(x + \color{blue}{\sqrt{16} \cdot \sqrt{x}}\right)}{-6}} \]
   metadata-eval [=>] 99.9	\[ \frac{1 - x}{\frac{1 + \left(x + \color{blue}{4} \cdot \sqrt{x}\right)}{-6}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7232

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

Alternative 2
Accuracy	95.3%
Cost	576

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

Alternative 3
Accuracy	95.4%
Cost	576

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

Alternative 4
Accuracy	95.4%
Cost	576

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

Alternative 5
Accuracy	95.4%
Cost	452

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

Alternative 6
Accuracy	95.3%
Cost	452

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

Alternative 7
Accuracy	95.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;-6 + x \cdot 12\\ \mathbf{else}:\\ \;\;\;\;6 + \frac{-12}{x}\\ \end{array} \]

Alternative 8
Accuracy	95.4%
Cost	196

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

Alternative 9
Accuracy	48.6%
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023133 
(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)))))