Average Error: 15.0 → 0.2
Time: 8.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r54462 = atan2(1.0, 0.0);
        double r54463 = 2.0;
        double r54464 = r54462 / r54463;
        double r54465 = 1.0;
        double r54466 = b;
        double r54467 = r54466 * r54466;
        double r54468 = a;
        double r54469 = r54468 * r54468;
        double r54470 = r54467 - r54469;
        double r54471 = r54465 / r54470;
        double r54472 = r54464 * r54471;
        double r54473 = r54465 / r54468;
        double r54474 = r54465 / r54466;
        double r54475 = r54473 - r54474;
        double r54476 = r54472 * r54475;
        return r54476;
}
↓
double f(double a, double b) {
        double r54477 = 0.5;
        double r54478 = atan2(1.0, 0.0);
        double r54479 = a;
        double r54480 = r54478 / r54479;
        double r54481 = b;
        double r54482 = r54478 / r54481;
        double r54483 = r54480 - r54482;
        double r54484 = r54477 * r54483;
        double r54485 = r54481 + r54479;
        double r54486 = r54484 / r54485;
        double r54487 = r54481 - r54479;
        double r54488 = r54486 / r54487;
        return r54488;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

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

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.4

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

  6. Applied associate-*r/9.3

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\pi}{a} - 0.5 \cdot \frac{\pi}{b}}}{b + a}}{b - a}\]

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 +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))))