Average Error: 13.6 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -24371.094079145096:\\ \;\;\;\;\frac{\frac{1}{F}}{\sin B \cdot F} - \left(\frac{1}{\sin B} + \frac{x}{\tan B}\right)\\ \mathbf{if}\;F \le 28156576218308.332:\\ \;\;\;\;(\left(\sqrt[3]{{\left({\left((F \cdot F + \left((2 \cdot x + 2)_*\right))_*\right)}^{\left(\frac{-1}{2}\right)}\right)}^{3}}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x}{\tan B}\right) - \frac{\frac{1}{F}}{\sin B \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -24371.094079145096

    1. Initial program 24.8

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

    2. Taylor expanded around -inf 12.2

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

    3. Applied simplify0.2

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

    if -24371.094079145096 < F < 28156576218308.332

    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

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

    4. Applied add-cbrt-cube0.3

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

    5. Applied simplify0.3

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

    if 28156576218308.332 < F

    1. Initial program 24.9

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

    2. Taylor expanded around inf 12.3

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

    3. Applied simplify0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018195 +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))))))