3frac (problem 3.3.3)

Percentage Accurate: 84.9% → 99.5%

Time: 12.5s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (/ -1.0 (* (+ 1.0 x) (* x (+ (* x -0.5) 0.5)))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	return -1.0 / ((1.0 + x) * (x * ((x * -0.5) + 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / ((1.0d0 + x) * (x * ((x * (-0.5d0)) + 0.5d0)))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

public static double code(double x) {
	return -1.0 / ((1.0 + x) * (x * ((x * -0.5) + 0.5)));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	return -1.0 / ((1.0 + x) * (x * ((x * -0.5) + 0.5)))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(-1.0 / Float64(Float64(1.0 + x) * Float64(x * Float64(Float64(x * -0.5) + 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = -1.0 / ((1.0 + x) * (x * ((x * -0.5) + 0.5)));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(-1.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(x * N[(N[(x * -0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	84.9%
Target	99.5%
Herbie	99.5%

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

Derivation

1. Initial program 83.2%

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

2. Simplified 83.2%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   Step-by-step derivation

   [Start] 83.2%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 83.2%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 83.2%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 83.2%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 83.2%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 83.2%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 83.2%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 83.2%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 83.2%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 83.2%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 60.5%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)}} \]
   Step-by-step derivation

   [Start] 83.2%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   clear-num [=>] 83.2%	\[ \frac{1}{1 + x} - \left(\color{blue}{\frac{1}{\frac{x}{2}}} - \frac{1}{x + -1}\right) \]
   frac-2neg [=>] 83.2%	\[ \frac{1}{1 + x} - \left(\frac{1}{\frac{x}{2}} - \color{blue}{\frac{-1}{-\left(x + -1\right)}}\right) \]
   metadata-eval [=>] 83.2%	\[ \frac{1}{1 + x} - \left(\frac{1}{\frac{x}{2}} - \frac{\color{blue}{-1}}{-\left(x + -1\right)}\right) \]
   frac-sub [=>] 60.5%	\[ \frac{1}{1 + x} - \color{blue}{\frac{1 \cdot \left(-\left(x + -1\right)\right) - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)}} \]
   *-un-lft-identity [<=] 60.5%	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(-\left(x + -1\right)\right)} - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   +-commutative [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(-\color{blue}{\left(-1 + x\right)}\right) - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   distribute-neg-in [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(\color{blue}{1} + \left(-x\right)\right) - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   sub-neg [<=] 60.5%	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(1 - x\right)} - \frac{x}{2} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   div-inv [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot \color{blue}{0.5}\right) \cdot -1}{\frac{x}{2} \cdot \left(-\left(x + -1\right)\right)} \]
   div-inv [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot \color{blue}{0.5}\right) \cdot \left(-\left(x + -1\right)\right)} \]
   +-commutative [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(-\color{blue}{\left(-1 + x\right)}\right)} \]
   distribute-neg-in [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)}} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(\color{blue}{1} + \left(-x\right)\right)} \]
   sub-neg [<=] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \color{blue}{\left(1 - x\right)}} \]

4. Simplified 60.5%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 + x \cdot -0.5\right)}} \]
   Step-by-step derivation

   [Start] 60.5%	\[ \frac{1}{1 + x} - \frac{\left(1 - x\right) - \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)} \]
   associate--l- [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{\color{blue}{1 - \left(x + \left(x \cdot 0.5\right) \cdot -1\right)}}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)} \]
   associate-*l* [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + \color{blue}{x \cdot \left(0.5 \cdot -1\right)}\right)}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot \color{blue}{-0.5}\right)}{\left(x \cdot 0.5\right) \cdot \left(1 - x\right)} \]
   associate-*l* [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{\color{blue}{x \cdot \left(0.5 \cdot \left(1 - x\right)\right)}} \]
   sub-neg [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}\right)} \]
   distribute-rgt-in [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \color{blue}{\left(1 \cdot 0.5 + \left(-x\right) \cdot 0.5\right)}} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(\color{blue}{0.5} + \left(-x\right) \cdot 0.5\right)} \]
   distribute-lft-neg-in [<=] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 + \color{blue}{\left(-x \cdot 0.5\right)}\right)} \]
   distribute-rgt-neg-in [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 + \color{blue}{x \cdot \left(-0.5\right)}\right)} \]
   metadata-eval [=>] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 + x \cdot \color{blue}{-0.5}\right)} \]

5. Applied egg-rr 62.0%

   \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot -0.5 + 0.5\right) - \left(1 + x\right) \cdot \left(1 - \left(x + x \cdot -0.5\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x \cdot -0.5 + 0.5\right)\right)}} \]
   Step-by-step derivation

   [Start] 60.5%	\[ \frac{1}{1 + x} - \frac{1 - \left(x + x \cdot -0.5\right)}{x \cdot \left(0.5 + x \cdot -0.5\right)} \]
   frac-sub [=>] 62.0%	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(0.5 + x \cdot -0.5\right)\right) - \left(1 + x\right) \cdot \left(1 - \left(x + x \cdot -0.5\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.5\right)\right)}} \]
   *-un-lft-identity [<=] 62.0%	\[ \frac{\color{blue}{x \cdot \left(0.5 + x \cdot -0.5\right)} - \left(1 + x\right) \cdot \left(1 - \left(x + x \cdot -0.5\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.5\right)\right)} \]
   +-commutative [=>] 62.0%	\[ \frac{x \cdot \color{blue}{\left(x \cdot -0.5 + 0.5\right)} - \left(1 + x\right) \cdot \left(1 - \left(x + x \cdot -0.5\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.5\right)\right)} \]
   +-commutative [=>] 62.0%	\[ \frac{x \cdot \left(x \cdot -0.5 + 0.5\right) - \left(1 + x\right) \cdot \left(1 - \left(x + x \cdot -0.5\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.5 + 0.5\right)}\right)} \]

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \frac{\color{blue}{-1}}{\left(1 + x\right) \cdot \left(x \cdot \left(x \cdot -0.5 + 0.5\right)\right)} \]

7. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	832

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

Alternative 2
Accuracy	84.1%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{1 + x} - \frac{1 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 4
Accuracy	75.6%
Cost	448

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

Alternative 5
Accuracy	52.2%
Cost	192

\[\frac{-2}{x} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))