Average Error: 13.8 → 0.2
Time: 5.0s
Precision: binary64
Cost: 704
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a}}{b \cdot a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\frac{\pi}{2}}{b + a}}{b \cdot 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 (/ (/ (/ PI 2.0) (+ 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 ((((double) M_PI) / 2.0) / (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
Error4.3
Cost704
\[\frac{\frac{\frac{\frac{\pi}{2}}{b + a}}{a}}{b}\]
Alternative 2
Error11.0
Cost904
\[\begin{array}{l} \mathbf{if}\;b \leq -1.092269238366496 \cdot 10^{-17} \lor \neg \left(b \leq 3.031150181682306 \cdot 10^{-28}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \end{array}\]
Alternative 3
Error19.5
Cost1218
\[\begin{array}{l} \mathbf{if}\;a \leq -2.0274635750868906 \cdot 10^{+75}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 6.2873135324240956 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 4
Error37.0
Cost64
\[0\]
Alternative 5
Error61.8
Cost64
\[1\]

Error

Derivation

  1. Initial program 13.8

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

  2. Simplified13.8

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

  4. Applied frac-sub_binary64_76913.8

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

  5. Applied associate-*r/_binary64_70213.8

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021044 
(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))))