Average Error: 19.5 → 0.5
Time: 24.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x + 1}}\right)} \cdot \frac{1}{\left|\sqrt[3]{x + 1}\right|}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x + 1}}\right)} \cdot \frac{1}{\left|\sqrt[3]{x + 1}\right|}
double f(double x) {
        double r7153478 = 1.0;
        double r7153479 = x;
        double r7153480 = sqrt(r7153479);
        double r7153481 = r7153478 / r7153480;
        double r7153482 = r7153479 + r7153478;
        double r7153483 = sqrt(r7153482);
        double r7153484 = r7153478 / r7153483;
        double r7153485 = r7153481 - r7153484;
        return r7153485;
}
↓
double f(double x) {
        double r7153486 = 1.0;
        double r7153487 = x;
        double r7153488 = r7153487 + r7153486;
        double r7153489 = sqrt(r7153488);
        double r7153490 = sqrt(r7153487);
        double r7153491 = r7153489 + r7153490;
        double r7153492 = cbrt(r7153488);
        double r7153493 = sqrt(r7153492);
        double r7153494 = r7153490 * r7153493;
        double r7153495 = r7153491 * r7153494;
        double r7153496 = r7153486 / r7153495;
        double r7153497 = fabs(r7153492);
        double r7153498 = r7153486 / r7153497;
        double r7153499 = r7153496 * r7153498;
        return r7153499;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.5
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.5

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

  3. Applied clear-num19.5

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

  4. Applied frac-sub19.5

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

  5. Simplified19.5

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

  6. Simplified19.5

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

  8. Applied flip--19.3

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

  9. Simplified18.9

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

  11. Applied add-cube-cbrt18.9

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

  12. Applied sqrt-prod18.9

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

  13. Applied associate-*l*18.9

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

  14. Applied *-un-lft-identity18.9

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

  15. Applied *-un-lft-identity18.9

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

  16. Applied times-frac18.9

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

  17. Applied times-frac18.9

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

  18. Simplified18.9

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

  19. Simplified0.5

    \[\leadsto \frac{1}{\left|\sqrt[3]{x + 1}\right|} \cdot \color{blue}{\frac{1 + 0}{\left(\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}\]

  20. Final simplification0.5

    \[\leadsto \frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x} \cdot \sqrt{\sqrt[3]{x + 1}}\right)} \cdot \frac{1}{\left|\sqrt[3]{x + 1}\right|}\]

Reproduce

herbie shell --seed 2019158 
(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)))))