Average Error: 14.1 → 0.2
Time: 19.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{b - a}{b + a} \cdot \pi}{a \cdot b}}{\left(b - a\right) \cdot 2}\]
\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{b - a}{b + a} \cdot \pi}{a \cdot b}}{\left(b - a\right) \cdot 2}
double f(double a, double b) {
        double r885297 = atan2(1.0, 0.0);
        double r885298 = 2.0;
        double r885299 = r885297 / r885298;
        double r885300 = 1.0;
        double r885301 = b;
        double r885302 = r885301 * r885301;
        double r885303 = a;
        double r885304 = r885303 * r885303;
        double r885305 = r885302 - r885304;
        double r885306 = r885300 / r885305;
        double r885307 = r885299 * r885306;
        double r885308 = r885300 / r885303;
        double r885309 = r885300 / r885301;
        double r885310 = r885308 - r885309;
        double r885311 = r885307 * r885310;
        return r885311;
}
↓
double f(double a, double b) {
        double r885312 = b;
        double r885313 = a;
        double r885314 = r885312 - r885313;
        double r885315 = r885312 + r885313;
        double r885316 = r885314 / r885315;
        double r885317 = atan2(1.0, 0.0);
        double r885318 = r885316 * r885317;
        double r885319 = r885313 * r885312;
        double r885320 = r885318 / r885319;
        double r885321 = 2.0;
        double r885322 = r885314 * r885321;
        double r885323 = r885320 / r885322;
        return r885323;
}

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

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

  4. Applied associate-*l/0.3

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

  6. Applied frac-sub0.3

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

  7. Applied associate-*r/0.3

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

  8. Simplified0.3

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

  10. Applied div-inv0.3

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

  11. Applied associate-*l*0.3

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

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