Average Error: 14.3 → 0.3
Time: 18.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \frac{\frac{1}{a \cdot b} \cdot 1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \frac{\frac{1}{a \cdot b} \cdot 1}{b + a}
double f(double a, double b) {
        double r46066 = atan2(1.0, 0.0);
        double r46067 = 2.0;
        double r46068 = r46066 / r46067;
        double r46069 = 1.0;
        double r46070 = b;
        double r46071 = r46070 * r46070;
        double r46072 = a;
        double r46073 = r46072 * r46072;
        double r46074 = r46071 - r46073;
        double r46075 = r46069 / r46074;
        double r46076 = r46068 * r46075;
        double r46077 = r46069 / r46072;
        double r46078 = r46069 / r46070;
        double r46079 = r46077 - r46078;
        double r46080 = r46076 * r46079;
        return r46080;
}
↓
double f(double a, double b) {
        double r46081 = atan2(1.0, 0.0);
        double r46082 = 2.0;
        double r46083 = r46081 / r46082;
        double r46084 = 1.0;
        double r46085 = a;
        double r46086 = b;
        double r46087 = r46085 * r46086;
        double r46088 = r46084 / r46087;
        double r46089 = r46088 * r46084;
        double r46090 = r46086 + r46085;
        double r46091 = r46089 / r46090;
        double r46092 = r46083 * r46091;
        return r46092;
}

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 14.3

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

  3. Applied associate-*l*14.3

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

  4. Simplified0.3

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

  5. Taylor expanded around 0 0.3

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

  7. Applied associate-*r/0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))