Average Error: 2.7 → 2.7
Time: 15.9s
Precision: 64
\[\frac{x}{y - z \cdot t}\]
↓
\[\frac{x}{\mathsf{fma}\left(-t, z, y\right)}\]
\frac{x}{y - z \cdot t}
↓
\frac{x}{\mathsf{fma}\left(-t, z, y\right)}
double f(double x, double y, double z, double t) {
        double r647054 = x;
        double r647055 = y;
        double r647056 = z;
        double r647057 = t;
        double r647058 = r647056 * r647057;
        double r647059 = r647055 - r647058;
        double r647060 = r647054 / r647059;
        return r647060;
}
↓
double f(double x, double y, double z, double t) {
        double r647061 = x;
        double r647062 = t;
        double r647063 = -r647062;
        double r647064 = z;
        double r647065 = y;
        double r647066 = fma(r647063, r647064, r647065);
        double r647067 = r647061 / r647066;
        return r647067;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.7
Target1.6
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.137830643487644440407921345820165445823 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

  1. Initial program 2.7

    \[\frac{x}{y - z \cdot t}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt3.3

    \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} - z \cdot t}\]

  4. Applied prod-diff8.7

    \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}}\]

  5. Simplified8.2

    \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-t, z, y\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)}\]

  6. Simplified2.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(-t, z, y\right) + \color{blue}{0}}\]

  7. Final simplification2.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(-t, z, y\right)}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e50) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 2.13783064348764444e131) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))