Average Error: 13.4 → 10.6
Time: 3.8m
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{{\left(\sqrt{\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}} \cdot \sqrt{\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}}\right)}^{\frac{-1}{2}}}{\frac{1}{F}} \cdot \frac{{\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{\tan B}\]

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. Simplified13.0

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

  4. Applied div-inv13.0

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

  5. Applied add-sqr-sqrt13.0

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

  6. Applied unpow-prod-down13.0

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

  7. Applied times-frac10.6

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

  9. Applied add-sqr-sqrt10.6

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

  10. Applied sqrt-prod10.6

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

  11. Final simplification10.6

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

Reproduce

herbie shell --seed 2019088 +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))))))