Average Error: 14.0 → 11.0
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{-x}{\tan B} + \frac{{\left(\sqrt{\left(x + x\right) + \left(2 + F \cdot F\right)}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot \frac{{\left(\sqrt{\left(x + x\right) + \left(2 + F \cdot F\right)}\right)}^{\left(-\frac{1}{2}\right)}}{\frac{1}{F}}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 14.0

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

  2. Applied simplify13.5

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

  4. Applied div-inv13.5

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

  5. Applied add-sqr-sqrt13.6

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

  6. Applied unpow-prod-down13.6

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

  7. Applied times-frac11.0

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

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(2479486159 2123901208 2662424940 349789437 14252662 202027171)' 
(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))))))