Average Error: 13.4 → 13.4
Time: 57.0s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[(\left({\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}}\right) \cdot \left(F \cdot \frac{1}{\sin B}\right) + \left(x \cdot \frac{-1}{\tan B}\right))_*\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.4

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

  2. Initial simplification13.3

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

  4. Applied div-inv13.3

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

  6. Applied div-inv13.4

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

  7. Final simplification13.4

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

Runtime

Time bar (total: 57.0s)Debug logProfile

herbie shell --seed 2018348 +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))))))