Average Error: 20.0 → 0.4
Time: 13.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt{x + 1} \cdot 1 + 1 \cdot \sqrt{x}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt{x + 1} \cdot 1 + 1 \cdot \sqrt{x}}}{\sqrt{x + 1}}
double f(double x) {
        double r8723467 = 1.0;
        double r8723468 = x;
        double r8723469 = sqrt(r8723468);
        double r8723470 = r8723467 / r8723469;
        double r8723471 = r8723468 + r8723467;
        double r8723472 = sqrt(r8723471);
        double r8723473 = r8723467 / r8723472;
        double r8723474 = r8723470 - r8723473;
        return r8723474;
}
↓
double f(double x) {
        double r8723475 = 1.0;
        double r8723476 = x;
        double r8723477 = sqrt(r8723476);
        double r8723478 = r8723475 / r8723477;
        double r8723479 = 1.0;
        double r8723480 = r8723476 + r8723479;
        double r8723481 = sqrt(r8723480);
        double r8723482 = r8723481 * r8723479;
        double r8723483 = r8723479 * r8723477;
        double r8723484 = r8723482 + r8723483;
        double r8723485 = r8723479 / r8723484;
        double r8723486 = r8723485 / r8723481;
        double r8723487 = r8723478 * r8723486;
        return r8723487;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied frac-sub20.0

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

  5. Applied flip--19.8

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

  6. Taylor expanded around 0 0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied times-frac0.4

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

  11. Applied times-frac0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))