Average Error: 9.5 → 0.1
Time: 5.0s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1} \]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\mathsf{fma}\left(x, x, x\right)}}{x - 1}

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ (/ 2.0 (fma x x x)) (- x 1.0)))
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 / fma(x, x, x)) / (x - 1.0);
}

Error

Bits error versus x

Target

Original9.5
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.5

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

  2. Applied frac-sub_binary6425.7

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

  3. Applied frac-add_binary6425.3

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

  4. Taylor expanded in x around 0 0.3

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

  5. Applied associate-/r*_binary640.1

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

  6. Simplified0.1

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

  7. Applied *-un-lft-identity_binary640.1

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

  8. Applied *-un-lft-identity_binary640.1

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

  9. Applied times-frac_binary640.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2022087 
(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))))