Average Error: 14.3 → 0.2
Time: 24.2min
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \left(\frac{\pi}{b + a} \cdot 1\right)}{2 \cdot \left(a \cdot b\right)}\]
\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 \left(\frac{\pi}{b + a} \cdot 1\right)}{2 \cdot \left(a \cdot b\right)}
double code(double a, double b) {
	return ((double) (((double) ((((double) M_PI) / 2.0) * (1.0 / ((double) (((double) (b * b)) - ((double) (a * a))))))) * ((double) ((1.0 / a) - (1.0 / b)))));
}
↓
double code(double a, double b) {
	return (((double) (1.0 * ((double) ((((double) M_PI) / ((double) (b + a))) * 1.0)))) / ((double) (2.0 * ((double) (a * b)))));
}

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 frac-sub14.3

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

  4. Applied associate-*l/14.3

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

  5. Applied frac-times14.3

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020181 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))