Average Error: 13.5 → 10.7
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{-x}{\tan B} + \frac{F}{\left(\sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} \cdot \sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\right) \cdot \sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.5

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

  2. Applied simplify13.4

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

  4. Applied pow-neg13.4

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

  5. Applied frac-times10.4

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

  6. Applied simplify10.4

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

  8. Applied add-cube-cbrt10.7

    \[\leadsto \frac{-x}{\tan B} + \frac{F}{\color{blue}{\left(\sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} \cdot \sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\right) \cdot \sqrt[3]{{\left(\left(x + x\right) + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}}}\]
  9. Removed slow pow expressions.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))))))