Average Error: 13.5 → 13.5
Time: 32.8s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[(\left({\left(\sqrt{\sqrt[3]{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}} \cdot \left|\sqrt[3]{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right|\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

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. Simplified13.4

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

  4. Applied add-sqr-sqrt13.5

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

  5. Applied unpow-prod-down13.5

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

  7. Applied add-cube-cbrt13.5

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

  8. Applied sqrt-prod13.5

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

  9. Simplified13.5

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

  10. Final simplification13.5

    \[\leadsto (\left({\left(\sqrt{\sqrt[3]{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}} \cdot \left|\sqrt[3]{(x \cdot 2 + \left((F \cdot F + 2)_*\right))_*}\right|\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

Reproduce

herbie shell --seed 2019030 +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))))))