3frac (problem 3.3.3)

Average Error: 15.17% → 0.1%

Time: 11.9s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{\frac{-1}{x \cdot 0.5}}{1 - x \cdot x} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ -1.0 (* x 0.5)) (- 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 (-1.0 / (x * 0.5)) / (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 = ((-1.0d0) / (x * 0.5d0)) / (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 (-1.0 / (x * 0.5)) / (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 (-1.0 / (x * 0.5)) / (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(-1.0 / Float64(x * 0.5)) / 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 = (-1.0 / (x * 0.5)) / (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[(-1.0 / N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	15.17%
Target	0.41%
Herbie	0.1%

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

Derivation

1. Initial program 15.17

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

2. Simplified 15.17

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

   [Start] 15.17	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 15.17	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 15.17	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 15.17	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 15.17	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 15.17	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 15.17	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 15.17	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 15.17	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 15.17	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 15.18

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

4. Applied egg-rr 40.57

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

5. Simplified 40.57

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

   [Start] 40.57	\[ \frac{-\left(x + x\right)}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
   neg-mul-1 [=>] 40.57	\[ \frac{\color{blue}{-1 \cdot \left(x + x\right)}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
   count-2 [=>] 40.57	\[ \frac{-1 \cdot \color{blue}{\left(2 \cdot x\right)}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
   associate-*r* [=>] 40.57	\[ \frac{\color{blue}{\left(-1 \cdot 2\right) \cdot x}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
   metadata-eval [=>] 40.57	\[ \frac{\color{blue}{-2} \cdot x}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
   neg-mul-1 [=>] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{-1 \cdot \mathsf{fma}\left(x, x, -1\right)}} + \left(-\frac{2}{x}\right) \]
   fma-udef [=>] 40.57	\[ \frac{-2 \cdot x}{-1 \cdot \color{blue}{\left(x \cdot x + -1\right)}} + \left(-\frac{2}{x}\right) \]
   distribute-lft-in [=>] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{-1 \cdot \left(x \cdot x\right) + -1 \cdot -1}} + \left(-\frac{2}{x}\right) \]
   associate-*l* [<=] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{\left(-1 \cdot x\right) \cdot x} + -1 \cdot -1} + \left(-\frac{2}{x}\right) \]
   neg-mul-1 [<=] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{\left(-x\right)} \cdot x + -1 \cdot -1} + \left(-\frac{2}{x}\right) \]
   metadata-eval [=>] 40.57	\[ \frac{-2 \cdot x}{\left(-x\right) \cdot x + \color{blue}{1}} + \left(-\frac{2}{x}\right) \]
   +-commutative [<=] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{1 + \left(-x\right) \cdot x}} + \left(-\frac{2}{x}\right) \]
   cancel-sign-sub-inv [<=] 40.57	\[ \frac{-2 \cdot x}{\color{blue}{1 - x \cdot x}} + \left(-\frac{2}{x}\right) \]

6. Applied egg-rr 40.08

   \[\leadsto \color{blue}{\frac{\left(\left(-2 \cdot x\right) \cdot \left(x \cdot 0.5\right) - 1\right) + x \cdot x}{\left(1 - x \cdot x\right) \cdot \left(x \cdot 0.5\right)}} \]

7. Simplified 40.08

   \[\leadsto \color{blue}{\frac{\frac{x \cdot x + \left(-1 - x \cdot x\right)}{x \cdot 0.5}}{1 - x \cdot x}} \]
   Proof

   [Start] 40.08	\[ \frac{\left(\left(-2 \cdot x\right) \cdot \left(x \cdot 0.5\right) - 1\right) + x \cdot x}{\left(1 - x \cdot x\right) \cdot \left(x \cdot 0.5\right)} \]
   *-commutative [=>] 40.08	\[ \frac{\left(\left(-2 \cdot x\right) \cdot \left(x \cdot 0.5\right) - 1\right) + x \cdot x}{\color{blue}{\left(x \cdot 0.5\right) \cdot \left(1 - x \cdot x\right)}} \]
   associate-/r* [=>] 40.08	\[ \color{blue}{\frac{\frac{\left(\left(-2 \cdot x\right) \cdot \left(x \cdot 0.5\right) - 1\right) + x \cdot x}{x \cdot 0.5}}{1 - x \cdot x}} \]

8. Taylor expanded in x around 0 0.1

   \[\leadsto \frac{\frac{\color{blue}{-1}}{x \cdot 0.5}}{1 - x \cdot x} \]

9. Final simplification 0.1

   \[\leadsto \frac{\frac{-1}{x \cdot 0.5}}{1 - x \cdot x} \]

Alternatives

Alternative 1
Error	0.41%
Cost	576

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

Alternative 2
Error	16.55%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	16.59%
Cost	448

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

Alternative 4
Error	96.72%
Cost	64

\[-1 \]

Alternative 5
Error	64.76%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023089 
(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))))