Average Error: 13.8 → 10.7
Time: 35.5s
Precision: 64
Internal precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left({F}^2 + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
⬇
\[\frac{-x}{\tan B} + \frac{F}{\sin B \cdot {\left(\left(x + x\right) + \left({F}^2 + 2\right)\right)}^{\left(\frac{1}{2}\right)}}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.8

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left({F}^2 + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

  2. Applied simplify 13.7

    \[\leadsto \color{blue}{\frac{-x}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(x + x\right) + \left({F}^2 + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}\]
  3. Using strategy rm

  4. Applied pow-neg 13.7

    \[\leadsto \frac{-x}{\tan B} + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{{\left(\left(x + x\right) + \left({F}^2 + 2\right)\right)}^{\left(\frac{1}{2}\right)}}}\]

  5. Applied frac-times 10.7

    \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{F \cdot 1}{\sin B \cdot {\left(\left(x + x\right) + \left({F}^2 + 2\right)\right)}^{\left(\frac{1}{2}\right)}}}\]

  6. Applied simplify 10.7

    \[\leadsto \frac{-x}{\tan B} + \frac{\color{blue}{F}}{\sin B \cdot {\left(\left(x + x\right) + \left({F}^2 + 2\right)\right)}^{\left(\frac{1}{2}\right)}}\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 35.5s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(4111036359 3348946013 2323121724 4064407186 3992456315 1464012481)'
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (sqr F) 2) (* 2 x)) (- (/ 1 2))))))