Average Error: 14.0 → 0.2
Time: 32.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -6436982318645.525390625:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 13740.11341628399168257601559162139892578:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]
\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 -6436982318645.525390625:\\
\;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\

\mathbf{elif}\;F \le 13740.11341628399168257601559162139892578:\\
\;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\

\end{array}
double f(double F, double B, double x) {
        double r44065 = x;
        double r44066 = 1.0;
        double r44067 = B;
        double r44068 = tan(r44067);
        double r44069 = r44066 / r44068;
        double r44070 = r44065 * r44069;
        double r44071 = -r44070;
        double r44072 = F;
        double r44073 = sin(r44067);
        double r44074 = r44072 / r44073;
        double r44075 = r44072 * r44072;
        double r44076 = 2.0;
        double r44077 = r44075 + r44076;
        double r44078 = r44076 * r44065;
        double r44079 = r44077 + r44078;
        double r44080 = r44066 / r44076;
        double r44081 = -r44080;
        double r44082 = pow(r44079, r44081);
        double r44083 = r44074 * r44082;
        double r44084 = r44071 + r44083;
        return r44084;
}
↓
double f(double F, double B, double x) {
        double r44085 = F;
        double r44086 = -6436982318645.525;
        bool r44087 = r44085 <= r44086;
        double r44088 = 1.0;
        double r44089 = B;
        double r44090 = sin(r44089);
        double r44091 = 2.0;
        double r44092 = pow(r44085, r44091);
        double r44093 = r44090 * r44092;
        double r44094 = r44088 / r44093;
        double r44095 = -1.0;
        double r44096 = r44095 / r44090;
        double r44097 = r44094 + r44096;
        double r44098 = x;
        double r44099 = r44098 * r44088;
        double r44100 = tan(r44089);
        double r44101 = r44099 / r44100;
        double r44102 = r44097 - r44101;
        double r44103 = 13740.113416283992;
        bool r44104 = r44085 <= r44103;
        double r44105 = r44085 / r44090;
        double r44106 = r44085 * r44085;
        double r44107 = 2.0;
        double r44108 = r44106 + r44107;
        double r44109 = r44107 * r44098;
        double r44110 = r44108 + r44109;
        double r44111 = r44088 / r44107;
        double r44112 = pow(r44110, r44111);
        double r44113 = r44105 / r44112;
        double r44114 = r44113 - r44101;
        double r44115 = 1.0;
        double r44116 = r44115 / r44090;
        double r44117 = r44116 - r44094;
        double r44118 = r44117 - r44101;
        double r44119 = r44104 ? r44114 : r44118;
        double r44120 = r44087 ? r44102 : r44119;
        return r44120;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -6436982318645.525

    1. Initial program 26.5

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

    2. Simplified26.5

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

    4. Applied associate-*r/26.4

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

    5. Taylor expanded around -inf 0.1

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

    6. Simplified0.1

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

    if -6436982318645.525 < F < 13740.113416283992

    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. Simplified0.4

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

    4. Applied associate-*r/0.3

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

    6. Applied pow-neg0.3

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

    7. Applied un-div-inv0.3

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

    if 13740.113416283992 < 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. Simplified25.0

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

    4. Applied associate-*r/24.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -6436982318645.525390625:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot {F}^{2}} + \frac{-1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 13740.11341628399168257601559162139892578:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))