Average Error: 14.3 → 0.2
Time: 1.4m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot \left(b - a\right)} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b}}{b - a}\]
\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{\frac{\pi}{2}}{a + b}}{a \cdot \left(b - a\right)} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b}}{b - a}
double f(double a, double b) {
        double r1683370 = atan2(1.0, 0.0);
        double r1683371 = 2.0;
        double r1683372 = r1683370 / r1683371;
        double r1683373 = 1.0;
        double r1683374 = b;
        double r1683375 = r1683374 * r1683374;
        double r1683376 = a;
        double r1683377 = r1683376 * r1683376;
        double r1683378 = r1683375 - r1683377;
        double r1683379 = r1683373 / r1683378;
        double r1683380 = r1683372 * r1683379;
        double r1683381 = r1683373 / r1683376;
        double r1683382 = r1683373 / r1683374;
        double r1683383 = r1683381 - r1683382;
        double r1683384 = r1683380 * r1683383;
        return r1683384;
}
↓
double f(double a, double b) {
        double r1683385 = atan2(1.0, 0.0);
        double r1683386 = 2.0;
        double r1683387 = r1683385 / r1683386;
        double r1683388 = a;
        double r1683389 = b;
        double r1683390 = r1683388 + r1683389;
        double r1683391 = r1683387 / r1683390;
        double r1683392 = r1683389 - r1683388;
        double r1683393 = r1683388 * r1683392;
        double r1683394 = r1683391 / r1683393;
        double r1683395 = r1683391 / r1683389;
        double r1683396 = r1683395 / r1683392;
        double r1683397 = r1683394 - r1683396;
        return r1683397;
}

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 14.3

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

  2. Simplified9.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{a} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{b}}\]
  3. Using strategy rm

  4. Applied div-inv9.1

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

  5. Applied associate-/l*4.6

    \[\leadsto \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{a} - \color{blue}{\frac{\frac{\frac{\pi}{2}}{a + b}}{\frac{b}{\frac{1}{b - a}}}}\]
  6. Using strategy rm

  7. Applied associate-/l/0.3

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

  9. Applied associate-/r/0.2

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

  10. Simplified0.2

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

  12. Applied associate-/r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))