Average Error: 10.0 → 0.3
Time: 4.4s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[2 \cdot \frac{1}{\mathsf{fma}\left(x, x, x\right) \cdot \left(x + -1\right)} \]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{1}{\mathsf{fma}\left(x, x, x\right) \cdot \left(x + -1\right)}

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

Error

Bits error versus x

Target

Original10.0
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 10.0

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
  2. Using strategy rm

  3. Applied frac-sub_binary6426.2

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

  4. Applied frac-add_binary6425.9

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

  5. Applied div-inv_binary6425.9

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

  6. Simplified25.9

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2021211 
(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))))