Average Error: 20.0 → 5.5
Time: 25.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1} \cdot \sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}
double f(double x) {
        double r6068095 = 1.0;
        double r6068096 = x;
        double r6068097 = sqrt(r6068096);
        double r6068098 = r6068095 / r6068097;
        double r6068099 = r6068096 + r6068095;
        double r6068100 = sqrt(r6068099);
        double r6068101 = r6068095 / r6068100;
        double r6068102 = r6068098 - r6068101;
        return r6068102;
}
↓
double f(double x) {
        double r6068103 = 1.0;
        double r6068104 = x;
        double r6068105 = r6068104 + r6068103;
        double r6068106 = sqrt(r6068105);
        double r6068107 = sqrt(r6068104);
        double r6068108 = r6068106 * r6068107;
        double r6068109 = r6068103 / r6068108;
        double r6068110 = r6068109 / r6068108;
        double r6068111 = r6068103 / r6068107;
        double r6068112 = r6068103 / r6068106;
        double r6068113 = r6068111 + r6068112;
        double r6068114 = r6068110 / r6068113;
        return r6068114;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie5.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

  3. Applied flip--20.0

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

  5. Applied associate-*r/22.5

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

  6. Applied associate-*l/20.0

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

  7. Applied frac-sub20.0

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

  9. Applied associate-*l/20.0

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

  10. Applied associate-*r/21.6

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

  11. Applied associate-*r/21.6

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

  12. Applied associate-*l/20.0

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

  13. Applied frac-sub19.8

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

  14. Simplified19.4

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

  15. Taylor expanded around 0 5.5

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

  16. Final simplification5.5

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

Reproduce

herbie shell --seed 2019171 +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)))))