Average Error: 14.5 → 0.3
Time: 6.6s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{a + b}}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{a + b}}{2 \cdot \left(b - a\right)}
(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))
↓
(FPCore (a b)
 :precision binary64
 (/ (/ (* PI (- (/ 1.0 a) (/ 1.0 b))) (+ a b)) (* 2.0 (- b a))))
double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}
↓
double code(double a, double b) {
	return ((((double) M_PI) * ((1.0 / a) - (1.0 / b))) / (a + b)) / (2.0 * (b - a));
}

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 14.5

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

  3. Applied difference-of-squares_binary649.5

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

  4. Applied add-cube-cbrt_binary649.5

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

  5. Applied times-frac_binary649.0

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

  6. Applied associate-*r*_binary649.0

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

  7. Simplified9.0

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

  9. Applied associate-*l/_binary649.0

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

  10. Applied frac-times_binary649.0

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

  11. Applied associate-*l/_binary640.3

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

  12. Simplified0.3

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

  14. Applied associate-*r/_binary640.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))