Average Error: 13.9 → 0.3
Time: 22.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot 1}{b - a}}{\frac{2 \cdot \left(b + a\right)}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi \cdot 1}{b - a}}{\frac{2 \cdot \left(b + a\right)}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r79160 = atan2(1.0, 0.0);
        double r79161 = 2.0;
        double r79162 = r79160 / r79161;
        double r79163 = 1.0;
        double r79164 = b;
        double r79165 = r79164 * r79164;
        double r79166 = a;
        double r79167 = r79166 * r79166;
        double r79168 = r79165 - r79167;
        double r79169 = r79163 / r79168;
        double r79170 = r79162 * r79169;
        double r79171 = r79163 / r79166;
        double r79172 = r79163 / r79164;
        double r79173 = r79171 - r79172;
        double r79174 = r79170 * r79173;
        return r79174;
}
↓
double f(double a, double b) {
        double r79175 = atan2(1.0, 0.0);
        double r79176 = 1.0;
        double r79177 = r79175 * r79176;
        double r79178 = b;
        double r79179 = a;
        double r79180 = r79178 - r79179;
        double r79181 = r79177 / r79180;
        double r79182 = 2.0;
        double r79183 = r79178 + r79179;
        double r79184 = r79182 * r79183;
        double r79185 = r79176 / r79179;
        double r79186 = r79176 / r79178;
        double r79187 = r79185 - r79186;
        double r79188 = r79184 / r79187;
        double r79189 = r79181 / r79188;
        return r79189;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.2

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  4. Applied add-sqr-sqrt9.2

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac8.8

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{\sqrt{1}}{b + a} \cdot \frac{\sqrt{1}}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied associate-*r*8.7

    \[\leadsto \color{blue}{\left(\left(\frac{\pi}{2} \cdot \frac{\sqrt{1}}{b + a}\right) \cdot \frac{\sqrt{1}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  7. Using strategy rm

  8. Applied frac-times8.7

    \[\leadsto \left(\color{blue}{\frac{\pi \cdot \sqrt{1}}{2 \cdot \left(b + a\right)}} \cdot \frac{\sqrt{1}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  9. Applied associate-*l/8.7

    \[\leadsto \color{blue}{\frac{\left(\pi \cdot \sqrt{1}\right) \cdot \frac{\sqrt{1}}{b - a}}{2 \cdot \left(b + a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  10. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \sqrt{1}\right) \cdot \frac{\sqrt{1}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b + a\right)}}\]

  11. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{\pi \cdot 1}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}{2 \cdot \left(b + a\right)}\]
  12. Using strategy rm

  13. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 1}{b - a}}{\frac{2 \cdot \left(b + a\right)}{\frac{1}{a} - \frac{1}{b}}}}\]

  14. Final simplification0.3

    \[\leadsto \frac{\frac{\pi \cdot 1}{b - a}}{\frac{2 \cdot \left(b + a\right)}{\frac{1}{a} - \frac{1}{b}}}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))