Average Error: 13.6 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.4042963914270186 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{\sin B}}{F \cdot F} - (x \cdot \left(\frac{\cos B}{\sin B}\right) + \left(\frac{1}{\sin B}\right))_*\\ \mathbf{if}\;F \le 5.514316409434656 \cdot 10^{+45}:\\ \;\;\;\;(\left({\left((F \cdot F + \left((2 \cdot x + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{\cos B}{\frac{\sin B}{-x}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{{F}^{2} \cdot \sin B} + \frac{\cos B \cdot x}{\sin B}\right)\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -1.4042963914270186e+17

    1. Initial program 25.1

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

    2. Applied simplify25.1

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

    4. Applied div-inv25.1

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

    5. Taylor expanded around inf 25.1

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

    6. Applied simplify25.1

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

    7. Taylor expanded around -inf 12.8

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

    8. Applied simplify0.2

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

    if -1.4042963914270186e+17 < F < 5.514316409434656e+45

    1. Initial program 0.5

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

    2. Applied simplify0.4

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

    4. Applied div-inv0.4

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

    5. Taylor expanded around inf 0.4

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

    6. Applied simplify0.5

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

    if 5.514316409434656e+45 < F

    1. Initial program 28.0

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

    2. Applied simplify28.0

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

    4. Applied div-inv28.0

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

    5. Taylor expanded around inf 28.0

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

    6. Applied simplify28.0

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

    7. Taylor expanded around inf 0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018193 +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))))))