Average Error: 13.4 → 10.7
Time: 37.1s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{{\left((2 \cdot x + \left((F \cdot F + 2)_*\right))_*\right)}^{\frac{-1}{2}} \cdot F}{\sin B} - \frac{x}{\sin B} \cdot \cos B\]

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. Simplified13.3

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

  4. Applied associate-*r/10.6

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

  6. Applied tan-quot10.7

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

  7. Applied associate-/r/10.7

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

  8. Final simplification10.7

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

Reproduce

herbie shell --seed 2019089 +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))))))