3frac (problem 3.3.3)

Average Error: 15.8% → 0.11%

Time: 9.2s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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

Target

Original	15.8%
Target	0.4%
Herbie	0.11%

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

Derivation

1. Initial program 15.8

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

2. Simplified 15.79

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

   [Start] 15.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 15.79	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 15.79	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 15.79	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 15.79	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 15.79	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 15.79	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 15.79	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 15.79	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 15.79	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 40.54

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

4. Simplified 40.55

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

   [Start] 40.54	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
   associate-/r* [=>] 40.55	\[ \color{blue}{\frac{\frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{1 + x}}{\mathsf{fma}\left(x, x, -x\right)}} \]

5. Taylor expanded in x around 0 0.11

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

6. Applied egg-rr 0.11

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

7. Final simplification 0.11

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

Alternatives

Alternative 1
Error	1.31%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-2 \cdot \frac{\frac{1}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \end{array} \]

Alternative 2
Error	16.67%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.64 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \end{array} \]

Alternative 3
Error	1.61%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{\left(x \cdot x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	1.31%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{1 + x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \end{array} \]

Alternative 5
Error	24.11%
Cost	713

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

Alternative 6
Error	0.41%
Cost	704

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

Alternative 7
Error	0.11%
Cost	704

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

Alternative 8
Error	24.78%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.16 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-2}{x}\\ \end{array} \]

Alternative 9
Error	48.57%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(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))))