Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 10.7s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

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

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 * (1.0 / (((-1.0 + (sqrt(x) * -4.0)) - x) / (x + -1.0)));
}

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

Python

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

↓

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

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

Target

Original	0.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.1

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

   [Start] 0.2	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \left(x - 1\right)} \]
   sub-neg [=>] 0.1	\[ \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   +-commutative [=>] 0.1	\[ \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \color{blue}{\left(\left(-1\right) + x\right)} \]
   distribute-rgt-in [=>] 0.1	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + x \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]
   *-commutative [=>] 0.1	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot x} \]
   cancel-sign-sub [<=] 0.1	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \left(-\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \cdot x} \]
   mul-1-neg [<=] 0.1	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \color{blue}{\left(-1 \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \cdot x \]
   metadata-eval [<=] 0.1	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \left(\color{blue}{\left(-1\right)} \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \cdot x \]
   *-commutative [=>] 0.1	\[ \left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} - \color{blue}{x \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]
   cancel-sign-sub-inv [=>] 0.1	\[ \color{blue}{\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} + \left(-x\right) \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]
   distribute-rgt1-in [=>] 0.1	\[ \color{blue}{\left(\left(-x\right) + 1\right) \cdot \left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right)} \]
   *-commutative [=>] 0.1	\[ \color{blue}{\left(\left(-1\right) \cdot \frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\right) \cdot \left(\left(-x\right) + 1\right)} \]
   *-commutative [=>] 0.1	\[ \color{blue}{\left(\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \left(-1\right)\right)} \cdot \left(\left(-x\right) + 1\right) \]
   associate-/r/ [<=] 0.1	\[ \color{blue}{\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{-1}}} \cdot \left(\left(-x\right) + 1\right) \]
   associate-/r/ [<=] 0.1	\[ \color{blue}{\frac{6}{\frac{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{-1}}{\left(-x\right) + 1}}} \]
   associate-/r* [<=] 0.1	\[ \frac{6}{\color{blue}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{\left(-1\right) \cdot \left(\left(-x\right) + 1\right)}}} \]
   associate-+l+ [=>] 0.1	\[ \frac{6}{\frac{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}}{\left(-1\right) \cdot \left(\left(-x\right) + 1\right)}} \]
   metadata-eval [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{\color{blue}{-1} \cdot \left(\left(-x\right) + 1\right)}} \]
   mul-1-neg [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{\color{blue}{-\left(\left(-x\right) + 1\right)}}} \]
   neg-sub0 [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{-\left(\color{blue}{\left(0 - x\right)} + 1\right)}} \]
   associate-+l- [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{-\color{blue}{\left(0 - \left(x - 1\right)\right)}}} \]
   sub0-neg [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{-\color{blue}{\left(-\left(x - 1\right)\right)}}} \]
   remove-double-neg [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{\color{blue}{x - 1}}} \]
   sub-neg [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{\color{blue}{x + \left(-1\right)}}} \]
   metadata-eval [=>] 0.1	\[ \frac{6}{\frac{x + \left(1 + 4 \cdot \sqrt{x}\right)}{x + \color{blue}{-1}}} \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{6}{\frac{\frac{1}{1 - \sqrt{x \cdot 16}} - \left(\frac{x \cdot 16}{1 - \sqrt{x \cdot 16}} - x\right)}{x + -1}} \]
   associate--r- [=>] 0.1	\[ \frac{6}{\frac{\color{blue}{\left(\frac{1}{1 - \sqrt{x \cdot 16}} - \frac{x \cdot 16}{1 - \sqrt{x \cdot 16}}\right) + x}}{x + -1}} \]
   div-sub [<=] 0.1	\[ \frac{6}{\frac{\color{blue}{\frac{1 - x \cdot 16}{1 - \sqrt{x \cdot 16}}} + x}{x + -1}} \]

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.7
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 14:\\ \;\;\;\;-6 \cdot \left(x + \left(1 + \sqrt{x} \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x} \cdot \left(\left(1 + x\right) - \sqrt{x \cdot 16}\right)\\ \end{array} \]

Alternative 2
Error	1.7
Cost	7236

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

Alternative 3
Error	0.1
Cost	7232

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

Alternative 4
Error	0.1
Cost	7232

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

Alternative 5
Error	3.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(x + -1\right) \cdot \left(6 + -6 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 6
Error	3.0
Cost	576

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

Alternative 7
Error	3.0
Cost	576

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

Alternative 8
Error	3.0
Cost	452

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

Alternative 9
Error	3.0
Cost	452

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

Alternative 10
Error	3.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left(x + -1\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{12}{x}\\ \end{array} \]

Alternative 11
Error	3.0
Cost	196

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

Alternative 12
Error	32.6
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2022354 
(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)))))