Average Error: 14.2 → 0.3
Time: 19.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(1 \cdot \frac{\frac{\pi}{2}}{a + b}\right)}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(1 \cdot \frac{\frac{\pi}{2}}{a + b}\right)}{b - a}
double f(double a, double b) {
        double r2646362 = atan2(1.0, 0.0);
        double r2646363 = 2.0;
        double r2646364 = r2646362 / r2646363;
        double r2646365 = 1.0;
        double r2646366 = b;
        double r2646367 = r2646366 * r2646366;
        double r2646368 = a;
        double r2646369 = r2646368 * r2646368;
        double r2646370 = r2646367 - r2646369;
        double r2646371 = r2646365 / r2646370;
        double r2646372 = r2646364 * r2646371;
        double r2646373 = r2646365 / r2646368;
        double r2646374 = r2646365 / r2646366;
        double r2646375 = r2646373 - r2646374;
        double r2646376 = r2646372 * r2646375;
        return r2646376;
}
↓
double f(double a, double b) {
        double r2646377 = 1.0;
        double r2646378 = a;
        double r2646379 = r2646377 / r2646378;
        double r2646380 = b;
        double r2646381 = r2646377 / r2646380;
        double r2646382 = r2646379 - r2646381;
        double r2646383 = atan2(1.0, 0.0);
        double r2646384 = 2.0;
        double r2646385 = r2646383 / r2646384;
        double r2646386 = r2646378 + r2646380;
        double r2646387 = r2646385 / r2646386;
        double r2646388 = r2646377 * r2646387;
        double r2646389 = r2646382 * r2646388;
        double r2646390 = r2646380 - r2646378;
        double r2646391 = r2646389 / r2646390;
        return r2646391;
}

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 14.2

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.0

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

  9. Applied associate-*r/9.0

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

  10. Applied associate-*l/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019192 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))