Average Error: 13.7 → 10.7
Time: 1.9m
Precision: 64
Internal Precision: 384
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{-x}{\tan B} + \left(\frac{1}{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} \cdot F\right) \cdot \frac{1}{\sin B}\]

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

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

  4. Applied div-inv13.6

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

  5. Applied associate-*r*10.6

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

  7. Applied pow-neg10.7

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(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))))))