3frac (problem 3.3.3)

Average Accuracy: 84.1% → 99.9%

Time: 9.3s

Precision: binary64

Cost: 704

Math

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

↓

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

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 -1.0))))

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 + -1.0));
}

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 + (-1.0d0)))
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 + -1.0));
}

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 + -1.0))

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

Target

Original	84.1%
Target	99.5%
Herbie	99.9%

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

Derivation

1. Initial program 84.1%

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

2. Simplified 84.1%

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

   [Start] 84.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 84.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 84.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 84.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 84.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 84.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 84.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 84.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 59.7%

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

   [Start] 84.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 59.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
   frac-sub [=>] 59.9	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}} \]
   *-un-lft-identity [<=] 59.9	\[ \frac{\color{blue}{x \cdot \left(x + -1\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   distribute-rgt-in [=>] 59.7	\[ \frac{\color{blue}{\left(x \cdot x + -1 \cdot x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   neg-mul-1 [<=] 59.7	\[ \frac{\left(x \cdot x + \color{blue}{\left(-x\right)}\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   sub-neg [<=] 59.7	\[ \frac{\color{blue}{\left(x \cdot x - x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   *-rgt-identity [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   distribute-rgt-in [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(x \cdot 2 + -1 \cdot 2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   metadata-eval [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   metadata-eval [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{\left(-2\right)}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   fma-def [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   metadata-eval [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
   distribute-rgt-in [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x + -1 \cdot x\right)}} \]
   neg-mul-1 [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x + \color{blue}{\left(-x\right)}\right)} \]
   sub-neg [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x - x\right)}} \]

4. Simplified 59.7%

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

   [Start] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   +-commutative [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \color{blue}{\left(x + 1\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   remove-double-neg [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(\color{blue}{\left(-\left(-x\right)\right)} + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   metadata-eval [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   distribute-neg-in [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\left(\left(-x\right) + -1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   neg-mul-1 [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   *-commutative [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{x \cdot -1} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   fma-udef [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\mathsf{fma}\left(x, -1, -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   distribute-lft-neg-in [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   distribute-lft-neg-in [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   fma-udef [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\left(x \cdot -1 + -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   *-commutative [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   neg-mul-1 [<=] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{\left(-x\right)} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   distribute-neg-in [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \color{blue}{\left(\left(-\left(-x\right)\right) + \left(--1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   remove-double-neg [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(\color{blue}{x} + \left(--1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   metadata-eval [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(x + \color{blue}{1}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   +-commutative [=>] 59.7	\[ \frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\color{blue}{\left(x + 1\right)} \cdot \left(x \cdot x - x\right)} \]

5. Taylor expanded in x around 0 99.5%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)}} \]
   Proof

   [Start] 99.5	\[ \frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)} \]
   add-log-exp [=>] 34.8	\[ \color{blue}{\log \left(e^{\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}}\right)} \]
   *-un-lft-identity [=>] 34.8	\[ \log \color{blue}{\left(1 \cdot e^{\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}}\right)} \]
   log-prod [=>] 34.8	\[ \color{blue}{\log 1 + \log \left(e^{\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}}\right)} \]
   metadata-eval [=>] 34.8	\[ \color{blue}{0} + \log \left(e^{\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}}\right) \]
   add-log-exp [<=] 99.5	\[ 0 + \color{blue}{\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}} \]
   associate-/r* [=>] 99.9	\[ 0 + \color{blue}{\frac{\frac{2}{x + 1}}{x \cdot x - x}} \]
   *-un-lft-identity [=>] 99.9	\[ 0 + \frac{\frac{2}{x + 1}}{x \cdot x - \color{blue}{1 \cdot x}} \]
   distribute-rgt-out-- [=>] 99.9	\[ 0 + \frac{\frac{2}{x + 1}}{\color{blue}{x \cdot \left(x - 1\right)}} \]
   sub-neg [=>] 99.9	\[ 0 + \frac{\frac{2}{x + 1}}{x \cdot \color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 99.9	\[ 0 + \frac{\frac{2}{x + 1}}{x \cdot \left(x + \color{blue}{-1}\right)} \]

7. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	832

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

Alternative 2
Accuracy	98.7%
Cost	777

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

Alternative 3
Accuracy	75.2%
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 4
Accuracy	75.6%
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 5
Accuracy	99.5%
Cost	576

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

Alternative 6
Accuracy	82.6%
Cost	448

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

Alternative 7
Accuracy	51.4%
Cost	192

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

Alternative 8
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023152 
(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))))