Average Error: 13.2 → 13.2
Time: 39.8s
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[(\left(\frac{1}{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\cos B \cdot \frac{-x}{\sin B}\right))_*\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.2

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

  2. Initial simplification13.1

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

  3. Taylor expanded around inf 13.1

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

  4. Simplified13.1

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

  6. Applied pow-neg13.2

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

  8. Applied *-un-lft-identity13.2

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

  9. Applied times-frac13.2

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

  10. Simplified13.2

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

  11. Final simplification13.2

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

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))