Average Error: 30.1 → 0.3
Time: 17.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\left(\sqrt{\sqrt{\sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} + \sqrt{x}}
double f(double x) {
        double r4914097 = x;
        double r4914098 = 1.0;
        double r4914099 = r4914097 + r4914098;
        double r4914100 = sqrt(r4914099);
        double r4914101 = sqrt(r4914097);
        double r4914102 = r4914100 - r4914101;
        return r4914102;
}
↓
double f(double x) {
        double r4914103 = 1.0;
        double r4914104 = x;
        double r4914105 = r4914104 + r4914103;
        double r4914106 = cbrt(r4914105);
        double r4914107 = sqrt(r4914106);
        double r4914108 = sqrt(r4914107);
        double r4914109 = sqrt(r4914105);
        double r4914110 = sqrt(r4914109);
        double r4914111 = r4914108 * r4914110;
        double r4914112 = r4914106 * r4914106;
        double r4914113 = sqrt(r4914112);
        double r4914114 = sqrt(r4914113);
        double r4914115 = r4914111 * r4914114;
        double r4914116 = sqrt(r4914104);
        double r4914117 = r4914115 + r4914116;
        double r4914118 = r4914103 / r4914117;
        return r4914118;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.9

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

  4. Simplified29.4

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

  5. Simplified29.4

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

  6. Taylor expanded around 0 0.2

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

  8. Applied add-sqr-sqrt0.2

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

  9. Applied sqrt-prod0.3

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

  11. Applied add-cube-cbrt0.3

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

  12. Applied sqrt-prod0.3

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

  13. Applied sqrt-prod0.3

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

  14. Applied associate-*l*0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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