Average Error: 30.3 → 0.2
Time: 34.4s
Precision: 64
\[\sqrt{x + 1.0} - \sqrt{x}\]
↓
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]
\sqrt{x + 1.0} - \sqrt{x}
↓
\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}
double f(double x) {
        double r20193556 = x;
        double r20193557 = 1.0;
        double r20193558 = r20193556 + r20193557;
        double r20193559 = sqrt(r20193558);
        double r20193560 = sqrt(r20193556);
        double r20193561 = r20193559 - r20193560;
        return r20193561;
}
↓
double f(double x) {
        double r20193562 = 1.0;
        double r20193563 = x;
        double r20193564 = r20193563 + r20193562;
        double r20193565 = sqrt(r20193564);
        double r20193566 = sqrt(r20193563);
        double r20193567 = r20193565 + r20193566;
        double r20193568 = r20193562 / r20193567;
        return r20193568;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  4. Simplified29.8

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

  5. Taylor expanded around 0 0.2

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

  6. Final simplification0.2

    \[\leadsto \frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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