Average Error: 9.5 → 0.3
Time: 21.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\sqrt{\sqrt{2}}}{\frac{x}{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\sqrt{\sqrt{2}}}{\frac{x}{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}
double f(double x) {
        double r4605108 = 1.0;
        double r4605109 = x;
        double r4605110 = r4605109 + r4605108;
        double r4605111 = r4605108 / r4605110;
        double r4605112 = 2.0;
        double r4605113 = r4605112 / r4605109;
        double r4605114 = r4605111 - r4605113;
        double r4605115 = r4605109 - r4605108;
        double r4605116 = r4605108 / r4605115;
        double r4605117 = r4605114 + r4605116;
        return r4605117;
}
↓
double f(double x) {
        double r4605118 = 2.0;
        double r4605119 = sqrt(r4605118);
        double r4605120 = sqrt(r4605119);
        double r4605121 = x;
        double r4605122 = cbrt(r4605119);
        double r4605123 = r4605121 / r4605122;
        double r4605124 = r4605120 / r4605123;
        double r4605125 = r4605121 * r4605121;
        double r4605126 = 1.0;
        double r4605127 = r4605125 - r4605126;
        double r4605128 = r4605122 * r4605122;
        double r4605129 = r4605127 / r4605128;
        double r4605130 = r4605120 / r4605129;
        double r4605131 = r4605124 * r4605130;
        return r4605131;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 9.5

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

  3. Applied frac-sub25.6

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

  4. Applied frac-add24.9

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

  5. Simplified25.3

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

  6. Taylor expanded around -inf 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt1.0

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

  9. Applied associate-/l*0.7

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

  10. Simplified0.7

    \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\left(x \cdot x - 1\right) \cdot x}{\sqrt{2}}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt0.7

    \[\leadsto \frac{\sqrt{2}}{\frac{\left(x \cdot x - 1\right) \cdot x}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}\]

  13. Applied times-frac0.8

    \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{x}{\sqrt[3]{\sqrt{2}}}}}\]

  14. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{x}{\sqrt[3]{\sqrt{2}}}}\]

  15. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{\sqrt{\sqrt{2}}}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{x}{\sqrt[3]{\sqrt{2}}}}}\]

  16. Final simplification0.3

    \[\leadsto \frac{\sqrt{\sqrt{2}}}{\frac{x}{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{x \cdot x - 1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}\]

Reproduce

herbie shell --seed 2019134 +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))))