Average Error: 9.9 → 0.3
Time: 2.9m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\mathsf{fma}\left(x, x, -1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\mathsf{fma}\left(x, x, -1\right) \cdot x}
double f(double x) {
        double r18301107 = 1.0;
        double r18301108 = x;
        double r18301109 = r18301108 + r18301107;
        double r18301110 = r18301107 / r18301109;
        double r18301111 = 2.0;
        double r18301112 = r18301111 / r18301108;
        double r18301113 = r18301110 - r18301112;
        double r18301114 = r18301108 - r18301107;
        double r18301115 = r18301107 / r18301114;
        double r18301116 = r18301113 + r18301115;
        return r18301116;
}
↓
double f(double x) {
        double r18301117 = 2.0;
        double r18301118 = x;
        double r18301119 = -1.0;
        double r18301120 = fma(r18301118, r18301118, r18301119);
        double r18301121 = r18301120 * r18301118;
        double r18301122 = r18301117 / r18301121;
        return r18301122;
}

Error

Bits error versus x

Target

Original9.9
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

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

  3. Applied frac-sub25.9

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

  4. Applied frac-add25.1

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

  5. Simplified25.5

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

  6. Simplified25.5

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

  7. Taylor expanded around -inf 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))