Average Error: 9.7 → 0.2
Time: 1.2m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\sqrt{\sqrt{2}}}{\frac{x - 1}{\sqrt{\sqrt{2}}}} \cdot \frac{\sqrt{2}}{\left(1 + x\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\sqrt{\sqrt{2}}}{\frac{x - 1}{\sqrt{\sqrt{2}}}} \cdot \frac{\sqrt{2}}{\left(1 + x\right) \cdot x}
double f(double x) {
        double r14137208 = 1.0;
        double r14137209 = x;
        double r14137210 = r14137209 + r14137208;
        double r14137211 = r14137208 / r14137210;
        double r14137212 = 2.0;
        double r14137213 = r14137212 / r14137209;
        double r14137214 = r14137211 - r14137213;
        double r14137215 = r14137209 - r14137208;
        double r14137216 = r14137208 / r14137215;
        double r14137217 = r14137214 + r14137216;
        return r14137217;
}
↓
double f(double x) {
        double r14137218 = 2.0;
        double r14137219 = sqrt(r14137218);
        double r14137220 = sqrt(r14137219);
        double r14137221 = x;
        double r14137222 = 1.0;
        double r14137223 = r14137221 - r14137222;
        double r14137224 = r14137223 / r14137220;
        double r14137225 = r14137220 / r14137224;
        double r14137226 = r14137222 + r14137221;
        double r14137227 = r14137226 * r14137221;
        double r14137228 = r14137219 / r14137227;
        double r14137229 = r14137225 * r14137228;
        return r14137229;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub25.8

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

  4. Applied frac-add25.0

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

  5. Taylor expanded around -inf 0.2

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

  7. Applied add-sqr-sqrt1.0

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

  8. Applied times-frac0.5

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied sqrt-prod0.2

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

  12. Applied associate-/l*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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