Average Error: 12.9 → 0.2
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -860605.7088315618:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 63806.85461578383:\\ \;\;\;\;\frac{\frac{F}{\frac{1}{{\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}}}}{\frac{\sin B}{{\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -860605.7088315618

    1. Initial program 22.5

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

    2. Initial simplification17.5

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

    3. Taylor expanded around -inf 0.2

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

    if -860605.7088315618 < F < 63806.85461578383

    1. Initial program 0.4

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

    2. Initial simplification0.3

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied unpow-prod-down0.3

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

    6. Applied *-un-lft-identity0.3

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

    7. Applied times-frac0.3

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

    8. Applied associate-/r*0.3

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

    if 63806.85461578383 < F

    1. Initial program 24.3

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

    2. Initial simplification18.6

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

    4. Applied add-sqr-sqrt18.6

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

    5. Applied unpow-prod-down18.7

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

    6. Applied *-un-lft-identity18.7

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

    7. Applied times-frac18.7

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

    8. Applied associate-/r*18.7

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

    9. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - \frac{1}{{F}^{2} \cdot \sin B}\right)} - \frac{x}{\tan B}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -860605.7088315618:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 63806.85461578383:\\ \;\;\;\;\frac{\frac{F}{\frac{1}{{\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}}}}{\frac{\sin B}{{\left(\sqrt{(2 \cdot x + \left((F \cdot F + 2)_*\right))_*}\right)}^{\frac{-1}{2}}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x}{\tan B}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018234 +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))))))