Average Error: 0.2 → 0.2
Time: 5.2s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[1 \cdot \frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
1 \cdot \frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}
double f(double B, double x) {
        double r8939 = x;
        double r8940 = 1.0;
        double r8941 = B;
        double r8942 = tan(r8941);
        double r8943 = r8940 / r8942;
        double r8944 = r8939 * r8943;
        double r8945 = -r8944;
        double r8946 = sin(r8941);
        double r8947 = r8940 / r8946;
        double r8948 = r8945 + r8947;
        return r8948;
}
↓
double f(double B, double x) {
        double r8949 = 1.0;
        double r8950 = B;
        double r8951 = cos(r8950);
        double r8952 = x;
        double r8953 = -r8952;
        double r8954 = 1.0;
        double r8955 = fma(r8951, r8953, r8954);
        double r8956 = sin(r8950);
        double r8957 = r8955 / r8956;
        double r8958 = r8949 * r8957;
        return r8958;
}

Error

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{1}{\sin B}\right)}\]
  3. Using strategy rm

  4. Applied tan-quot0.2

    \[\leadsto \mathsf{fma}\left(-x, \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}}, \frac{1}{\sin B}\right)\]

  5. Applied associate-/r/0.2

    \[\leadsto \mathsf{fma}\left(-x, \color{blue}{\frac{1}{\sin B} \cdot \cos B}, \frac{1}{\sin B}\right)\]

  6. Taylor expanded around inf 0.2

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\sin B} - 1 \cdot \frac{x \cdot \cos B}{\sin B}}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(\cos B \cdot \left(-x\right) + 1\right)}\]
  8. Using strategy rm

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}}\]

  12. Final simplification0.2

    \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))