Average Error: 13.5 → 0.7
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -2.2926544526369904 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(F \cdot F\right) \cdot \sin B} - (\left(\frac{\cos B}{\sin B}\right) \cdot x + \left(\frac{1}{\sin B}\right))_*\\ \mathbf{if}\;F \le 30372589.530890018:\\ \;\;\;\;(\left({\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\sin B} \cdot \cos B\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{{F}^{2} \cdot \sin B}\right) - \frac{\cos B}{\frac{\sin B}{x}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -2.2926544526369904e+113

    1. Initial program 34.6

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

    2. Applied simplify34.6

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

    4. Applied pow-neg34.6

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

    5. Taylor expanded around inf 34.6

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

    6. Applied simplify34.6

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

    7. Taylor expanded around -inf 0.2

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

    8. Applied simplify0.2

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

    if -2.2926544526369904e+113 < F < 30372589.530890018

    1. Initial program 1.1

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

    2. Applied simplify1.0

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

    4. Applied tan-quot1.0

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

    5. Applied associate-/r/1.0

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

    if 30372589.530890018 < F

    1. Initial program 24.4

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

    2. Applied simplify24.4

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

    4. Applied pow-neg24.4

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

    5. Taylor expanded around inf 24.4

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

    6. Applied simplify24.4

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

    7. Taylor expanded around inf 0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +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))))))