Average Error: 14.4 → 0.3
Time: 38.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{a + b}}{2}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\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}{a + b}}{2}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r2688065 = atan2(1.0, 0.0);
        double r2688066 = 2.0;
        double r2688067 = r2688065 / r2688066;
        double r2688068 = 1.0;
        double r2688069 = b;
        double r2688070 = r2688069 * r2688069;
        double r2688071 = a;
        double r2688072 = r2688071 * r2688071;
        double r2688073 = r2688070 - r2688072;
        double r2688074 = r2688068 / r2688073;
        double r2688075 = r2688067 * r2688074;
        double r2688076 = r2688068 / r2688071;
        double r2688077 = r2688068 / r2688069;
        double r2688078 = r2688076 - r2688077;
        double r2688079 = r2688075 * r2688078;
        return r2688079;
}
↓
double f(double a, double b) {
        double r2688080 = 1.0;
        double r2688081 = atan2(1.0, 0.0);
        double r2688082 = a;
        double r2688083 = b;
        double r2688084 = r2688082 + r2688083;
        double r2688085 = r2688081 / r2688084;
        double r2688086 = 2.0;
        double r2688087 = r2688085 / r2688086;
        double r2688088 = r2688080 * r2688087;
        double r2688089 = r2688083 - r2688082;
        double r2688090 = r2688080 / r2688082;
        double r2688091 = r2688080 / r2688083;
        double r2688092 = r2688090 - r2688091;
        double r2688093 = r2688089 / r2688092;
        double r2688094 = r2688088 / r2688093;
        return r2688094;
}

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 14.4

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified9.0

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

  9. Applied associate-*r/8.9

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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))))