3frac (problem 3.3.3)

Average Error: 9.9 → 0.3

Time: 11.1s

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	9.9
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 9.9

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

2. Simplified 9.9

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

   [Start] 9.9	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 9.9	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.9	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 9.9	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 9.9	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 9.9	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 9.9	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 9.9	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.9	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 9.9	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 10.0

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

4. Simplified 10.0

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

   [Start] 10.0	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
   *-commutative [=>] 10.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
   /-rgt-identity [=>] 10.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
   *-commutative [=>] 10.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(\color{blue}{x \cdot 2} - x\right)}{x} \]

5. Applied egg-rr 25.8

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

6. Simplified 25.8

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

   [Start] 25.8	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(x + -2\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
   associate-/r* [=>] 25.8	\[ \color{blue}{\frac{\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(x + -2\right)}{1 + x}}{x \cdot x - x}} \]
   sub-neg [=>] 25.8	\[ \frac{\frac{\color{blue}{\left(x \cdot x - x\right) + \left(-\left(1 + x\right) \cdot \left(x + -2\right)\right)}}{1 + x}}{x \cdot x - x} \]
   distribute-rgt-neg-in [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \color{blue}{\left(1 + x\right) \cdot \left(-\left(x + -2\right)\right)}}{1 + x}}{x \cdot x - x} \]
   +-commutative [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \color{blue}{\left(x + 1\right)} \cdot \left(-\left(x + -2\right)\right)}{1 + x}}{x \cdot x - x} \]
   +-commutative [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \left(-\color{blue}{\left(-2 + x\right)}\right)}{1 + x}}{x \cdot x - x} \]
   distribute-neg-in [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \color{blue}{\left(\left(--2\right) + \left(-x\right)\right)}}{1 + x}}{x \cdot x - x} \]
   metadata-eval [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \left(\color{blue}{2} + \left(-x\right)\right)}{1 + x}}{x \cdot x - x} \]
   sub-neg [<=] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \color{blue}{\left(2 - x\right)}}{1 + x}}{x \cdot x - x} \]
   +-commutative [=>] 25.8	\[ \frac{\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \left(2 - x\right)}{\color{blue}{x + 1}}}{x \cdot x - x} \]

7. Taylor expanded in x around 0 0.1

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

8. Applied egg-rr 0.1

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

9. Simplified 0.3

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

   [Start] 0.1	\[ -\frac{\frac{2}{x + 1}}{-\left(x \cdot x - x\right)} \]
   associate-/l/ [=>] 0.3	\[ -\color{blue}{\frac{2}{\left(-\left(x \cdot x - x\right)\right) \cdot \left(x + 1\right)}} \]
   distribute-neg-frac [=>] 0.3	\[ \color{blue}{\frac{-2}{\left(-\left(x \cdot x - x\right)\right) \cdot \left(x + 1\right)}} \]
   metadata-eval [=>] 0.3	\[ \frac{\color{blue}{-2}}{\left(-\left(x \cdot x - x\right)\right) \cdot \left(x + 1\right)} \]
   *-commutative [=>] 0.3	\[ \frac{-2}{\color{blue}{\left(x + 1\right) \cdot \left(-\left(x \cdot x - x\right)\right)}} \]
   neg-sub0 [=>] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \color{blue}{\left(0 - \left(x \cdot x - x\right)\right)}} \]
   sub-neg [=>] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \left(0 - \color{blue}{\left(x \cdot x + \left(-x\right)\right)}\right)} \]
   +-commutative [<=] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \left(0 - \color{blue}{\left(\left(-x\right) + x \cdot x\right)}\right)} \]
   associate--r+ [=>] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \color{blue}{\left(\left(0 - \left(-x\right)\right) - x \cdot x\right)}} \]
   neg-sub0 [<=] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \left(\color{blue}{\left(-\left(-x\right)\right)} - x \cdot x\right)} \]
   remove-double-neg [=>] 0.3	\[ \frac{-2}{\left(x + 1\right) \cdot \left(\color{blue}{x} - x \cdot x\right)} \]

10. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	713

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

Alternative 2
Error	15.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x\\ \end{array} \]

Alternative 3
Error	10.8
Cost	448

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

Alternative 4
Error	30.9
Cost	192

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

Alternative 5
Error	61.9
Cost	64

\[-1 \]

Alternative 6
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023059 
(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))))