Average Error: 13.4 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -6.325313922069097 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{x}{F}}{\sin B \cdot F} - (\left(\frac{\cos B}{\sin B}\right) \cdot x + \left(\frac{1}{\sin B}\right))_*\\ \mathbf{if}\;F \le 106122265.98117906:\\ \;\;\;\;(\left({\left(\sqrt{(F \cdot F + \left((2 \cdot x + 2)_*\right))_*}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{(F \cdot F + \left((2 \cdot x + 2)_*\right))_*}\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - (\left(\frac{\cos B}{\sin B}\right) \cdot x + \left(\frac{\frac{x}{F}}{\sin B \cdot F}\right))_*\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -6.325313922069097e+23

    1. Initial program 25.3

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

    2. Applied simplify25.3

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

    3. Taylor expanded around -inf 0.2

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

    4. Applied simplify0.2

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

    if -6.325313922069097e+23 < F < 106122265.98117906

    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. Applied simplify0.3

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

    4. Applied add-sqr-sqrt0.4

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

    5. Applied unpow-prod-down0.4

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

    if 106122265.98117906 < F

    1. Initial program 25.0

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

    2. Applied simplify25.0

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

    3. Taylor expanded around inf 0.2

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

    4. Applied simplify0.2

      \[\leadsto \color{blue}{\frac{1}{\sin B} - (\left(\frac{\cos B}{\sin B}\right) \cdot x + \left(\frac{\frac{x}{F}}{\sin B \cdot F}\right))_*}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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))))))