Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Accuracy: 99.7% → 99.9%

Time: 9.8s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (/ (- 1.0 x) (+ (* (sqrt x) -4.0) (- -1.0 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) / ((sqrt(x) * -4.0) + (-1.0 - 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) / ((sqrt(x) * (-4.0d0)) + ((-1.0d0) - 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) / ((Math.sqrt(x) * -4.0) + (-1.0 - 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) / ((math.sqrt(x) * -4.0) + (-1.0 - 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(Float64(1.0 - x) / Float64(Float64(sqrt(x) * -4.0) + Float64(-1.0 - 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) / ((sqrt(x) * -4.0) + (-1.0 - 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[(N[(1.0 - x), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0), $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 99.8%

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

2. Simplified 99.9%

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

   [Start] 99.8	\[ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 99.6%

   \[\leadsto \color{blue}{\frac{x + -1}{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot 0.16666666666666666}} \]
   Step-by-step derivation

   [Start] 99.9	\[ \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \cdot \left(x + -1\right) \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(x + -1\right) \cdot \frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]
   clear-num [=>] 99.7	\[ \left(x + -1\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{6}}} \]
   un-div-inv [=>] 99.9	\[ \color{blue}{\frac{x + -1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}{6}}} \]
   div-inv [=>] 99.6	\[ \frac{x + -1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right) \cdot \frac{1}{6}}} \]
   fma-udef [=>] 99.6	\[ \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)} \cdot \frac{1}{6}} \]
   associate-+r+ [=>] 99.6	\[ \frac{x + -1}{\color{blue}{\left(\left(4 \cdot \sqrt{x} + x\right) + 1\right)} \cdot \frac{1}{6}} \]
   fma-def [=>] 99.6	\[ \frac{x + -1}{\left(\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1\right) \cdot \frac{1}{6}} \]
   metadata-eval [=>] 99.6	\[ \frac{x + -1}{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot \color{blue}{0.16666666666666666}} \]

4. Simplified 99.9%

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

   [Start] 99.6	\[ \frac{x + -1}{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot 0.16666666666666666} \]
   /-rgt-identity [<=] 99.6	\[ \frac{x + -1}{\color{blue}{\frac{\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right) \cdot 0.16666666666666666}{1}}} \]
   associate-/l* [=>] 99.9	\[ \frac{x + -1}{\color{blue}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}{\frac{1}{0.16666666666666666}}}} \]
   metadata-eval [=>] 99.9	\[ \frac{x + -1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}{\color{blue}{6}}} \]
   associate-/r/ [=>] 100.0	\[ \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \cdot 6} \]
   *-lft-identity [<=] 100.0	\[ \frac{x + -1}{\color{blue}{1 \cdot \left(\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1\right)}} \cdot 6 \]
   *-lft-identity [=>] 100.0	\[ \frac{x + -1}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \cdot 6 \]
   fma-udef [=>] 100.0	\[ \frac{x + -1}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right)} + 1} \cdot 6 \]
   +-commutative [=>] 100.0	\[ \frac{x + -1}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right)} + 1} \cdot 6 \]
   associate-+r+ [<=] 99.9	\[ \frac{x + -1}{\color{blue}{x + \left(4 \cdot \sqrt{x} + 1\right)}} \cdot 6 \]
   fma-udef [<=] 99.9	\[ \frac{x + -1}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \cdot 6 \]

5. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6 \]
   frac-2neg [=>] 99.9	\[ \color{blue}{\frac{-\left(x + -1\right)}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)}} \cdot 6 \]
   div-inv [=>] 99.9	\[ \color{blue}{\left(\left(-\left(x + -1\right)\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)}\right)} \cdot 6 \]
   +-commutative [=>] 99.9	\[ \left(\left(-\color{blue}{\left(-1 + x\right)}\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)}\right) \cdot 6 \]
   distribute-neg-in [=>] 99.9	\[ \left(\color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)}\right) \cdot 6 \]
   metadata-eval [=>] 99.9	\[ \left(\left(\color{blue}{1} + \left(-x\right)\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)}\right) \cdot 6 \]

6. Simplified 99.9%

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

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

7. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	7232

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

Alternative 2
Accuracy	95.4%
Cost	6848

\[\frac{\mathsf{fma}\left(6, x, -6\right)}{1 + x} \]

Alternative 3
Accuracy	95.6%
Cost	452

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

Alternative 4
Accuracy	95.6%
Cost	452

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

Alternative 5
Accuracy	95.6%
Cost	196

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

Alternative 6
Accuracy	48.4%
Cost	64

\[-6 \]

Reproduce

herbie shell --seed 2023157 
(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)))))