Average Error: 14.1 → 0.3
Time: 39.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r2547388 = atan2(1.0, 0.0);
        double r2547389 = 2.0;
        double r2547390 = r2547388 / r2547389;
        double r2547391 = 1.0;
        double r2547392 = b;
        double r2547393 = r2547392 * r2547392;
        double r2547394 = a;
        double r2547395 = r2547394 * r2547394;
        double r2547396 = r2547393 - r2547395;
        double r2547397 = r2547391 / r2547396;
        double r2547398 = r2547390 * r2547397;
        double r2547399 = r2547391 / r2547394;
        double r2547400 = r2547391 / r2547392;
        double r2547401 = r2547399 - r2547400;
        double r2547402 = r2547398 * r2547401;
        return r2547402;
}
↓
double f(double a, double b) {
        double r2547403 = 1.0;
        double r2547404 = atan2(1.0, 0.0);
        double r2547405 = 2.0;
        double r2547406 = r2547404 / r2547405;
        double r2547407 = a;
        double r2547408 = b;
        double r2547409 = r2547407 + r2547408;
        double r2547410 = r2547406 / r2547409;
        double r2547411 = r2547403 * r2547410;
        double r2547412 = r2547408 - r2547407;
        double r2547413 = r2547403 / r2547407;
        double r2547414 = r2547403 / r2547408;
        double r2547415 = r2547413 - r2547414;
        double r2547416 = r2547412 / r2547415;
        double r2547417 = r2547411 / r2547416;
        return r2547417;
}

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 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. Using strategy rm

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.7

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

  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. Using strategy rm

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))