Average Error: 20.0 → 0.4
Time: 20.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{1 + x}}}{1 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
1 \cdot \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{1 + x}}}{1 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}
double f(double x) {
        double r6307035 = 1.0;
        double r6307036 = x;
        double r6307037 = sqrt(r6307036);
        double r6307038 = r6307035 / r6307037;
        double r6307039 = r6307036 + r6307035;
        double r6307040 = sqrt(r6307039);
        double r6307041 = r6307035 / r6307040;
        double r6307042 = r6307038 - r6307041;
        return r6307042;
}
↓
double f(double x) {
        double r6307043 = 1.0;
        double r6307044 = 1.0;
        double r6307045 = x;
        double r6307046 = sqrt(r6307045);
        double r6307047 = r6307044 / r6307046;
        double r6307048 = r6307043 + r6307045;
        double r6307049 = sqrt(r6307048);
        double r6307050 = r6307047 / r6307049;
        double r6307051 = r6307046 + r6307049;
        double r6307052 = r6307043 * r6307051;
        double r6307053 = r6307050 / r6307052;
        double r6307054 = r6307043 * r6307053;
        return r6307054;
}

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. Applied associate-/l/19.8

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

  7. Taylor expanded around 0 0.8

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

  9. Applied div-inv0.8

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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)))))