Average Error: 14.1 → 0.3
Time: 1.6m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{b + a} \cdot \frac{\pi}{2}}{a \cdot \left(b - a\right)} - \frac{\frac{\pi}{2}}{b + a} \cdot \frac{\frac{1}{b - a}}{b}\]
\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{1}{b + a} \cdot \frac{\pi}{2}}{a \cdot \left(b - a\right)} - \frac{\frac{\pi}{2}}{b + a} \cdot \frac{\frac{1}{b - a}}{b}
double f(double a, double b) {
        double r1898369 = atan2(1.0, 0.0);
        double r1898370 = 2.0;
        double r1898371 = r1898369 / r1898370;
        double r1898372 = 1.0;
        double r1898373 = b;
        double r1898374 = r1898373 * r1898373;
        double r1898375 = a;
        double r1898376 = r1898375 * r1898375;
        double r1898377 = r1898374 - r1898376;
        double r1898378 = r1898372 / r1898377;
        double r1898379 = r1898371 * r1898378;
        double r1898380 = r1898372 / r1898375;
        double r1898381 = r1898372 / r1898373;
        double r1898382 = r1898380 - r1898381;
        double r1898383 = r1898379 * r1898382;
        return r1898383;
}
↓
double f(double a, double b) {
        double r1898384 = 1.0;
        double r1898385 = b;
        double r1898386 = a;
        double r1898387 = r1898385 + r1898386;
        double r1898388 = r1898384 / r1898387;
        double r1898389 = atan2(1.0, 0.0);
        double r1898390 = 2.0;
        double r1898391 = r1898389 / r1898390;
        double r1898392 = r1898388 * r1898391;
        double r1898393 = r1898385 - r1898386;
        double r1898394 = r1898386 * r1898393;
        double r1898395 = r1898392 / r1898394;
        double r1898396 = r1898391 / r1898387;
        double r1898397 = r1898384 / r1898393;
        double r1898398 = r1898397 / r1898385;
        double r1898399 = r1898396 * r1898398;
        double r1898400 = r1898395 - r1898399;
        return r1898400;
}

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 14.1

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

  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 *-un-lft-identity9.0

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

  5. Applied div-inv9.0

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

  6. Applied times-frac4.6

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

  8. Applied associate-/l/0.3

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

  10. Applied div-inv0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))