Average Error: 0.2 → 0.6
Time: 10.6s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\left(-\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{1}{\tan B}\right)\right) + \frac{1}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\left(-\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{1}{\tan B}\right)\right) + \frac{1}{\sin B}
double f(double B, double x) {
        double r18661 = x;
        double r18662 = 1.0;
        double r18663 = B;
        double r18664 = tan(r18663);
        double r18665 = r18662 / r18664;
        double r18666 = r18661 * r18665;
        double r18667 = -r18666;
        double r18668 = sin(r18663);
        double r18669 = r18662 / r18668;
        double r18670 = r18667 + r18669;
        return r18670;
}
↓
double f(double B, double x) {
        double r18671 = x;
        double r18672 = cbrt(r18671);
        double r18673 = r18672 * r18672;
        double r18674 = 1.0;
        double r18675 = B;
        double r18676 = tan(r18675);
        double r18677 = r18674 / r18676;
        double r18678 = r18672 * r18677;
        double r18679 = r18673 * r18678;
        double r18680 = -r18679;
        double r18681 = sin(r18675);
        double r18682 = r18674 / r18681;
        double r18683 = r18680 + r18682;
        return r18683;
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.6

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

  4. Applied associate-*l*0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))