3frac (problem 3.3.3)

Average Accuracy: 84.8% → 99.9%

Time: 12.5s

Precision: binary64

Cost: 576

Math

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

↓

\[\frac{\frac{-2}{x}}{1 - x \cdot x} \]

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))))

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));
}

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))
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));
}

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))

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

Target

Original	84.8%
Target	99.6%
Herbie	99.9%

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

Derivation

1. Initial program 84.8%

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

2. Simplified 84.8%

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

   [Start] 84.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 84.8	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.8	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 84.8	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 84.8	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 84.8	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 84.8	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 84.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 84.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 59.8%

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

   [Start] 84.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 59.3	\[ \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
   frac-sub [=>] 59.9	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}} \]

4. Simplified 59.8%

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

   [Start] 59.8	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   +-commutative [=>] 59.8	\[ \frac{x \cdot x - \left(x + \color{blue}{\left(x + 1\right)} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   +-commutative [=>] 59.8	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   associate-+l- [=>] 59.8	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   *-commutative [=>] 59.8	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   +-commutative [=>] 59.8	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\color{blue}{\left(x + 1\right)} \cdot \mathsf{fma}\left(x, x, -x\right)} \]

5. Taylor expanded in x around 0 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(\frac{x}{\frac{2}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{-1}} \]
   Proof

   [Start] 99.6	\[ \frac{2}{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   clear-num [=>] 99.6	\[ \color{blue}{\frac{1}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, -x\right)}{2}}} \]
   inv-pow [=>] 99.6	\[ \color{blue}{{\left(\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, -x\right)}{2}\right)}^{-1}} \]
   *-commutative [=>] 99.6	\[ {\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, -x\right) \cdot \left(x + 1\right)}}{2}\right)}^{-1} \]
   associate-/l* [=>] 99.5	\[ {\color{blue}{\left(\frac{\mathsf{fma}\left(x, x, -x\right)}{\frac{2}{x + 1}}\right)}}^{-1} \]
   fma-udef [=>] 99.5	\[ {\left(\frac{\color{blue}{x \cdot x + \left(-x\right)}}{\frac{2}{x + 1}}\right)}^{-1} \]
   neg-mul-1 [=>] 99.5	\[ {\left(\frac{x \cdot x + \color{blue}{-1 \cdot x}}{\frac{2}{x + 1}}\right)}^{-1} \]
   distribute-rgt-out [=>] 99.5	\[ {\left(\frac{\color{blue}{x \cdot \left(x + -1\right)}}{\frac{2}{x + 1}}\right)}^{-1} \]
   metadata-eval [<=] 99.5	\[ {\left(\frac{x \cdot \left(x + \color{blue}{\left(-1\right)}\right)}{\frac{2}{x + 1}}\right)}^{-1} \]
   sub-neg [<=] 99.5	\[ {\left(\frac{x \cdot \color{blue}{\left(x - 1\right)}}{\frac{2}{x + 1}}\right)}^{-1} \]
   associate-/l* [=>] 99.5	\[ {\color{blue}{\left(\frac{x}{\frac{\frac{2}{x + 1}}{x - 1}}\right)}}^{-1} \]
   associate-/r* [<=] 99.6	\[ {\left(\frac{x}{\color{blue}{\frac{2}{\left(x + 1\right) \cdot \left(x - 1\right)}}}\right)}^{-1} \]
   difference-of-sqr-1 [<=] 99.6	\[ {\left(\frac{x}{\frac{2}{\color{blue}{x \cdot x - 1}}}\right)}^{-1} \]
   fma-neg [=>] 99.6	\[ {\left(\frac{x}{\frac{2}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}}\right)}^{-1} \]
   metadata-eval [=>] 99.6	\[ {\left(\frac{x}{\frac{2}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}}\right)}^{-1} \]

7. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{2}{\mathsf{fma}\left(x, x, -1\right)}} \]
   Proof

   [Start] 99.6	\[ {\left(\frac{x}{\frac{2}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{-1} \]
   unpow-1 [=>] 99.6	\[ \color{blue}{\frac{1}{\frac{x}{\frac{2}{\mathsf{fma}\left(x, x, -1\right)}}}} \]
   associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{1}{x} \cdot \frac{2}{\mathsf{fma}\left(x, x, -1\right)}} \]

8. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{1}{x} \cdot \frac{2}{\mathsf{fma}\left(x, x, -1\right)} \]
   associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{\frac{1}{x} \cdot 2}{\mathsf{fma}\left(x, x, -1\right)}} \]
   frac-2neg [=>] 99.9	\[ \color{blue}{\frac{-\frac{1}{x} \cdot 2}{-\mathsf{fma}\left(x, x, -1\right)}} \]
   associate-*l/ [=>] 99.9	\[ \frac{-\color{blue}{\frac{1 \cdot 2}{x}}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   metadata-eval [=>] 99.9	\[ \frac{-\frac{\color{blue}{2}}{x}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   distribute-neg-frac [=>] 99.9	\[ \frac{\color{blue}{\frac{-2}{x}}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{\color{blue}{-2}}{x}}{-\mathsf{fma}\left(x, x, -1\right)} \]
   neg-sub0 [=>] 99.9	\[ \frac{\frac{-2}{x}}{\color{blue}{0 - \mathsf{fma}\left(x, x, -1\right)}} \]
   metadata-eval [<=] 99.9	\[ \frac{\frac{-2}{x}}{\color{blue}{\log 1} - \mathsf{fma}\left(x, x, -1\right)} \]
   fma-udef [=>] 99.9	\[ \frac{\frac{-2}{x}}{\log 1 - \color{blue}{\left(x \cdot x + -1\right)}} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{-2}{x}}{\log 1 - \color{blue}{\left(-1 + x \cdot x\right)}} \]
   associate--r+ [=>] 99.9	\[ \frac{\frac{-2}{x}}{\color{blue}{\left(\log 1 - -1\right) - x \cdot x}} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{-2}{x}}{\left(\color{blue}{0} - -1\right) - x \cdot x} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{-2}{x}}{\color{blue}{1} - x \cdot x} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	841

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

Alternative 2
Accuracy	83.5%
Cost	713

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

Alternative 3
Accuracy	76.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.37 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 4
Accuracy	52.0%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))