Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Accuracy: 99.7% → 99.9%

Time: 7.3s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ (+ x (* 4.0 (sqrt x))) 1.0) (+ 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 / (((x + (4.0 * sqrt(x))) + 1.0) / (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 / (((x + (4.0d0 * sqrt(x))) + 1.0d0) / (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 / (((x + (4.0 * Math.sqrt(x))) + 1.0) / (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 / (((x + (4.0 * math.sqrt(x))) + 1.0) / (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(Float64(Float64(x + Float64(4.0 * sqrt(x))) + 1.0) / 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 / (((x + (4.0 * sqrt(x))) + 1.0) / (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[(N[(N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $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. Simplified 99.9%

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

   [Start] 99.7	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   associate-*l/ [<=] 99.9	\[ \color{blue}{\frac{6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot \left(x - 1\right)} \]
   +-commutative [=>] 99.9	\[ \frac{6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \cdot \left(x - 1\right) \]
   fma-def [=>] 99.9	\[ \frac{6}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \cdot \left(x - 1\right) \]
   sub-neg [=>] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   metadata-eval [=>] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \left(x + \color{blue}{-1}\right) \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \left(x + -1\right) \]
   associate-*l/ [=>] 99.7	\[ \color{blue}{\frac{6 \cdot \left(x + -1\right)}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{x + -1}}} \]
   fma-udef [=>] 99.9	\[ \frac{6}{\frac{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}}{x + -1}} \]
   associate-+r+ [=>] 99.9	\[ \frac{6}{\frac{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}}{x + -1}} \]
   fma-def [=>] 99.9	\[ \frac{6}{\frac{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1}{x + -1}} \]

4. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}{x + -1}} \]
   fma-udef [=>] 99.9	\[ \frac{6}{\frac{\color{blue}{\left(4 \cdot \sqrt{x} + x\right)} + 1}{x + -1}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	7232

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

Alternative 2
Accuracy	95.4%
Cost	704

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

Alternative 3
Accuracy	95.4%
Cost	576

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

Alternative 4
Accuracy	95.4%
Cost	452

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

Alternative 5
Accuracy	95.4%
Cost	452

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

Alternative 6
Accuracy	95.4%
Cost	452

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

Alternative 7
Accuracy	95.4%
Cost	196

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

Alternative 8
Accuracy	48.4%
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023151 
(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)))))