Average Error: 13.7 → 10.7
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\left(\left({\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot F\right) \cdot \frac{1}{\sin B} + \frac{-x}{\tan B}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.7

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

  2. Applied simplify13.6

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

  4. Applied add-sqr-sqrt13.6

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

  5. Applied unpow-prod-down13.6

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

  7. Applied fma-udef13.6

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

  9. Applied div-inv13.6

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

  10. Applied associate-*r*10.7

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018214 +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))))))