Average Error: 19.6 → 0.4
Time: 14.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \left(\frac{1}{\sqrt{x}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right)\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \left(\frac{1}{\sqrt{x}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right)
double f(double x) {
        double r2016567 = 1.0;
        double r2016568 = x;
        double r2016569 = sqrt(r2016568);
        double r2016570 = r2016567 / r2016569;
        double r2016571 = r2016568 + r2016567;
        double r2016572 = sqrt(r2016571);
        double r2016573 = r2016567 / r2016572;
        double r2016574 = r2016570 - r2016573;
        return r2016574;
}
↓
double f(double x) {
        double r2016575 = 1.0;
        double r2016576 = x;
        double r2016577 = r2016576 + r2016575;
        double r2016578 = sqrt(r2016577);
        double r2016579 = sqrt(r2016576);
        double r2016580 = r2016578 + r2016579;
        double r2016581 = r2016575 / r2016580;
        double r2016582 = r2016581 / r2016578;
        double r2016583 = sqrt(r2016582);
        double r2016584 = r2016575 / r2016579;
        double r2016585 = r2016584 * r2016583;
        double r2016586 = r2016583 * r2016585;
        return r2016586;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original19.6
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied frac-sub19.5

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

  4. Simplified19.5

    \[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  5. Using strategy rm

  6. Applied flip--19.3

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

  7. Simplified18.9

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

  9. Applied div-inv18.9

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

  10. Applied times-frac18.9

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

  11. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 + 0}{\sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{1 + 0}{\sqrt{x}} \cdot \color{blue}{\left(\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right)}\]

  14. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\frac{1 + 0}{\sqrt{x}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right) \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}}\]

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))