3frac (problem 3.3.3)

Percentage Accurate: 84.1% → 99.6%

Time: 10.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = 2.0d0 / ((1.0d0 + x) * ((x * x) - x))
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 2.0 / ((1.0 + x) * ((x * x) - x));
}

Python

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

↓

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

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(2.0 / Float64(Float64(1.0 + x) * Float64(Float64(x * x) - x)))
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 = 2.0 / ((1.0 + x) * ((x * x) - x));
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[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	84.1%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 85.1%

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

2. Simplified 85.1%

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

   [Start] 85.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 85.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 85.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 85.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 85.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 85.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 85.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 85.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 85.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 85.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 59.1%

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

   [Start] 85.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   flip-+ [=>] 61.0	\[ \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   sub-neg [=>] 61.0	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   metadata-eval [<=] 61.0	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   distribute-neg-in [<=] 61.0	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   +-commutative [<=] 61.0	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   associate-/r/ [=>] 59.1	\[ \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   metadata-eval [=>] 59.1	\[ \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   +-commutative [=>] 59.1	\[ \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   distribute-neg-in [=>] 59.1	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   metadata-eval [=>] 59.1	\[ \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   sub-neg [<=] 59.1	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

4. Simplified 59.8%

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

   [Start] 59.1	\[ \frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   associate-*l/ [=>] 59.8	\[ \color{blue}{\frac{1 \cdot \left(1 - x\right)}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   *-lft-identity [=>] 59.8	\[ \frac{\color{blue}{1 - x}}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

5. Applied egg-rr 60.7%

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

   [Start] 59.8	\[ \frac{1 - x}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   clear-num [=>] 61.0	\[ \color{blue}{\frac{1}{\frac{1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 83.9	\[ \frac{1}{\frac{1 - x \cdot x}{1 - x}} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
   frac-sub [=>] 60.7	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)}} \]
   *-un-lft-identity [<=] 60.7	\[ \frac{\color{blue}{x \cdot \left(x + -1\right)} - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   metadata-eval [<=] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \frac{\color{blue}{1 \cdot 1} - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   flip-+ [<=] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \color{blue}{\left(1 + x\right)} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   *-rgt-identity [=>] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   +-commutative [=>] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(2 \cdot \color{blue}{\left(-1 + x\right)} - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   distribute-lft-in [=>] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(2 \cdot -1 + 2 \cdot x\right)} - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   metadata-eval [=>] 60.7	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(\left(\color{blue}{-2} + 2 \cdot x\right) - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]

6. Taylor expanded in x around 0 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

8. Simplified 99.9%

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

   [Start] 99.9	\[ \frac{2}{\left(1 + x\right) \cdot \left(-1 \cdot x + {x}^{2}\right)} \]
   neg-mul-1 [<=] 99.9	\[ \frac{2}{\left(1 + x\right) \cdot \left(\color{blue}{\left(-x\right)} + {x}^{2}\right)} \]
   unpow2 [=>] 99.9	\[ \frac{2}{\left(1 + x\right) \cdot \left(\left(-x\right) + \color{blue}{x \cdot x}\right)} \]
   +-commutative [<=] 99.9	\[ \frac{2}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x + \left(-x\right)\right)}} \]
   sub-neg [<=] 99.9	\[ \frac{2}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x - x\right)}} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	713

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

Alternative 2
Accuracy	99.6%
Cost	704

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

Alternative 3
Accuracy	51.9%
Cost	192

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

Alternative 4
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023160 
(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))))