Average Error: 29.6 → 0.2
Time: 10.6s
Precision: 64
\[\sqrt{x + 1.0} - \sqrt{x}\]
↓
\[\frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}\]
\sqrt{x + 1.0} - \sqrt{x}
↓
\frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}
double f(double x) {
        double r8355199 = x;
        double r8355200 = 1.0;
        double r8355201 = r8355199 + r8355200;
        double r8355202 = sqrt(r8355201);
        double r8355203 = sqrt(r8355199);
        double r8355204 = r8355202 - r8355203;
        return r8355204;
}
↓
double f(double x) {
        double r8355205 = 1.0;
        double r8355206 = x;
        double r8355207 = sqrt(r8355206);
        double r8355208 = sqrt(r8355205);
        double r8355209 = hypot(r8355207, r8355208);
        double r8355210 = r8355207 + r8355209;
        double r8355211 = r8355205 / r8355210;
        return r8355211;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

    \[\sqrt{x + 1.0} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.4

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1.0} \cdot \sqrt{x + 1.0} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1.0} + \sqrt{x}}}\]

  4. Simplified28.9

    \[\leadsto \frac{\color{blue}{\left(x + 1.0\right) - x}}{\sqrt{x + 1.0} + \sqrt{x}}\]

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{1.0}}{\sqrt{x + 1.0} + \sqrt{x}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1.0}{\sqrt{x + \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}} + \sqrt{x}}\]

  8. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1.0}{\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{1.0} \cdot \sqrt{1.0}} + \sqrt{x}}\]

  9. Applied hypot-def0.2

    \[\leadsto \frac{1.0}{\color{blue}{\mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)} + \sqrt{x}}\]

  10. Final simplification0.2

    \[\leadsto \frac{1.0}{\sqrt{x} + \mathsf{hypot}\left(\sqrt{x}, \sqrt{1.0}\right)}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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