Average Error: 1.8 → 1.9
Time: 16.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)\right)}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r57534 = 1.0;
        double r57535 = 2.0;
        double r57536 = r57534 / r57535;
        double r57537 = l;
        double r57538 = r57535 * r57537;
        double r57539 = Om;
        double r57540 = r57538 / r57539;
        double r57541 = pow(r57540, r57535);
        double r57542 = kx;
        double r57543 = sin(r57542);
        double r57544 = pow(r57543, r57535);
        double r57545 = ky;
        double r57546 = sin(r57545);
        double r57547 = pow(r57546, r57535);
        double r57548 = r57544 + r57547;
        double r57549 = r57541 * r57548;
        double r57550 = r57534 + r57549;
        double r57551 = sqrt(r57550);
        double r57552 = r57534 / r57551;
        double r57553 = r57534 + r57552;
        double r57554 = r57536 * r57553;
        double r57555 = sqrt(r57554);
        return r57555;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r57556 = 1.0;
        double r57557 = 2.0;
        double r57558 = r57556 / r57557;
        double r57559 = l;
        double r57560 = r57557 * r57559;
        double r57561 = Om;
        double r57562 = r57560 / r57561;
        double r57563 = pow(r57562, r57557);
        double r57564 = kx;
        double r57565 = sin(r57564);
        double r57566 = pow(r57565, r57557);
        double r57567 = ky;
        double r57568 = sin(r57567);
        double r57569 = pow(r57568, r57557);
        double r57570 = r57566 + r57569;
        double r57571 = r57563 * r57570;
        double r57572 = expm1(r57571);
        double r57573 = log1p(r57572);
        double r57574 = r57556 + r57573;
        double r57575 = sqrt(r57574);
        double r57576 = r57556 / r57575;
        double r57577 = r57556 + r57576;
        double r57578 = r57558 * r57577;
        double r57579 = sqrt(r57578);
        return r57579;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied log1p-expm1-u1.9

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)\right)}}}\right)}\]

  4. Final simplification1.9

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)\right)}}\right)}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))