3frac (problem 3.3.3)

Average Accuracy: 84.6% → 99.6%

Time: 10.7s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ -2.0 (* (+ x 1.0) (- 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 / ((x + 1.0) * (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) / ((x + 1.0d0) * (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 / ((x + 1.0) * (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 / ((x + 1.0) * (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(x + 1.0) * Float64(x - Float64(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 / ((x + 1.0) * (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[(x + 1.0), $MachinePrecision] * N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	84.6%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 86.5%

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

2. Simplified 86.5%

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

   [Start] 86.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 86.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 86.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 86.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 86.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 86.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 86.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 86.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 86.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 86.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 60.8%

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

   [Start] 86.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-2neg [=>] 86.5	\[ \frac{1}{1 + x} - \left(\color{blue}{\frac{-2}{-x}} - \frac{1}{x + -1}\right) \]
   frac-2neg [=>] 86.5	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \color{blue}{\frac{-1}{-\left(x + -1\right)}}\right) \]
   metadata-eval [=>] 86.5	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \frac{\color{blue}{-1}}{-\left(x + -1\right)}\right) \]
   frac-sub [=>] 60.8	\[ \frac{1}{1 + x} - \color{blue}{\frac{\left(-2\right) \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)}} \]
   metadata-eval [=>] 60.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2} \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
   +-commutative [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
   distribute-neg-in [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
   metadata-eval [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
   sub-neg [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(1 - x\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
   +-commutative [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\color{blue}{\left(-1 + x\right)}\right)} \]
   distribute-neg-in [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)}} \]
   metadata-eval [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(\color{blue}{1} + \left(-x\right)\right)} \]
   sub-neg [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(1 - x\right)}} \]

4. Simplified 60.8%

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

   [Start] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(1 - x\right)} \]
   cancel-sign-sub [=>] 60.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) + x \cdot -1}}{\left(-x\right) \cdot \left(1 - x\right)} \]
   *-commutative [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{-1 \cdot x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
   neg-mul-1 [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{\left(-x\right)}}{\left(-x\right) \cdot \left(1 - x\right)} \]
   unsub-neg [=>] 60.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) - x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
   sub-neg [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)}} \]
   +-commutative [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(\left(-x\right) + 1\right)}} \]
   distribute-lft-in [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{\left(-x\right) \cdot \left(-x\right) + \left(-x\right) \cdot 1}} \]
   sqr-neg [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} + \left(-x\right) \cdot 1} \]
   unpow2 [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2}} + \left(-x\right) \cdot 1} \]
   *-rgt-identity [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{{x}^{2} + \color{blue}{\left(-x\right)}} \]
   sub-neg [<=] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2} - x}} \]
   unpow2 [=>] 60.8	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} - x} \]

5. Taylor expanded in x around 0 52.8%

   \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{-2}}{x \cdot x - x} \]

6. Applied egg-rr 60.4%

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

   [Start] 52.8	\[ \frac{1}{1 + x} - \frac{-2}{x \cdot x - x} \]
   frac-2neg [=>] 52.8	\[ \frac{1}{1 + x} - \color{blue}{\frac{--2}{-\left(x \cdot x - x\right)}} \]
   metadata-eval [=>] 52.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{2}}{-\left(x \cdot x - x\right)} \]
   frac-sub [=>] 60.4	\[ \color{blue}{\frac{1 \cdot \left(-\left(x \cdot x - x\right)\right) - \left(1 + x\right) \cdot 2}{\left(1 + x\right) \cdot \left(-\left(x \cdot x - x\right)\right)}} \]
   *-un-lft-identity [<=] 60.4	\[ \frac{\color{blue}{\left(-\left(x \cdot x - x\right)\right)} - \left(1 + x\right) \cdot 2}{\left(1 + x\right) \cdot \left(-\left(x \cdot x - x\right)\right)} \]
   +-commutative [=>] 60.4	\[ \frac{\left(-\left(x \cdot x - x\right)\right) - \color{blue}{\left(x + 1\right)} \cdot 2}{\left(1 + x\right) \cdot \left(-\left(x \cdot x - x\right)\right)} \]
   +-commutative [=>] 60.4	\[ \frac{\left(-\left(x \cdot x - x\right)\right) - \left(x + 1\right) \cdot 2}{\color{blue}{\left(x + 1\right)} \cdot \left(-\left(x \cdot x - x\right)\right)} \]

7. Taylor expanded in x around 0 99.6%

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

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	76.2%
Cost	585

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

Alternative 2
Accuracy	76.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	448

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

Alternative 4
Accuracy	51.6%
Cost	192

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

Alternative 5
Accuracy	3.3%
Cost	64

\[-1 \]

Alternative 6
Accuracy	3.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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))))