Average Error: 14.5 → 0.3
Time: 4.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}
double f(double a, double b) {
        double r36187 = atan2(1.0, 0.0);
        double r36188 = 2.0;
        double r36189 = r36187 / r36188;
        double r36190 = 1.0;
        double r36191 = b;
        double r36192 = r36191 * r36191;
        double r36193 = a;
        double r36194 = r36193 * r36193;
        double r36195 = r36192 - r36194;
        double r36196 = r36190 / r36195;
        double r36197 = r36189 * r36196;
        double r36198 = r36190 / r36193;
        double r36199 = r36190 / r36191;
        double r36200 = r36198 - r36199;
        double r36201 = r36197 * r36200;
        return r36201;
}
↓
double f(double a, double b) {
        double r36202 = 0.5;
        double r36203 = atan2(1.0, 0.0);
        double r36204 = a;
        double r36205 = b;
        double r36206 = r36204 * r36205;
        double r36207 = r36203 / r36206;
        double r36208 = r36202 * r36207;
        double r36209 = 1.0;
        double r36210 = r36205 + r36204;
        double r36211 = r36209 / r36210;
        double r36212 = r36208 * r36211;
        return r36212;
}

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 14.5

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

  3. Applied difference-of-squares9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.2

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

  7. Simplified9.2

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

  9. Applied associate-*l/9.2

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]
  12. Using strategy rm

  13. Applied div-inv0.3

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}}\]

  14. Final simplification0.3

    \[\leadsto \left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))