Average Error: 14.5 → 0.2
Time: 9.6s
Precision: binary64
Cost: 704
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b + a}
(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 (/ (* 0.5 (/ PI (* b a))) (+ 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 (0.5 * (((double) M_PI) / (b * a))) / (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

Alternatives

Alternative 1
Error0.3
Cost832
\[\left(0.5 \cdot \frac{\pi}{b \cdot a}\right) \cdot \frac{1}{b + a}\]
Alternative 2
Error0.3
Cost1088
\[\frac{\frac{0.5}{b - a} \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}\]
Alternative 3
Error0.3
Cost1216
\[\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b + a}\]
Alternative 4
Error0.3
Cost1216
\[\frac{\frac{\frac{\pi}{2} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{b - a}\]
Alternative 5
Error0.3
Cost1216
\[\frac{\frac{\frac{\pi}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}\]
Alternative 6
Error0.3
Cost1344
\[\frac{1}{b + a} \cdot \frac{\left(b - a\right) \cdot \frac{\frac{\pi}{2}}{b - a}}{b \cdot a}\]
Alternative 7
Error0.5
Cost14144
\[\frac{\left(\left(\pi \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{0.5}}{b - a}}{b + a}\]
Alternative 8
Error0.5
Cost14272
\[\frac{\frac{1}{\sqrt{2}}}{b + a} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\frac{\pi}{\sqrt{2}}}{b - a}\right)\]
Alternative 9
Error0.7
Cost14272
\[\frac{\sqrt{\frac{\pi}{2}}}{b + a} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\sqrt{\frac{\pi}{2}}}{b - a}\right)\]
Alternative 10
Error0.8
Cost22592
\[\frac{1}{b + a} \cdot \left(\sqrt[3]{\frac{\frac{\pi}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \cdot \left(\sqrt[3]{\frac{\frac{\pi}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \cdot \sqrt[3]{\frac{\frac{\pi}{2}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}\right)\right)\]
Alternative 11
Error0.9
Cost40384
\[\frac{\left(\left(\pi \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \cdot \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{\sqrt[3]{b - a} \cdot \sqrt[3]{b - a}}\right) \cdot \frac{\sqrt[3]{0.5}}{\sqrt[3]{b - a}}}{b + a}\]

Error

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. Simplified14.5

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

  4. Applied difference-of-squares_binary64_7299.7

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

  5. Applied *-un-lft-identity_binary64_7609.7

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

  6. Applied times-frac_binary64_7669.2

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

  7. Applied associate-*l*_binary64_7010.3

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

  8. Simplified0.3

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

  10. Applied associate-*l/_binary64_7030.3

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

  11. Simplified0.3

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

  12. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]

  13. Simplified0.2

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b + a}}\]

  14. Final simplification0.2

    \[\leadsto \frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b + a}\]

Reproduce

herbie shell --seed 2021043 
(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))))