\[\frac{x}{y - z \cdot t}\]

↓

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

\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 r725878 = x;
        double r725879 = y;
        double r725880 = z;
        double r725881 = t;
        double r725882 = r725880 * r725881;
        double r725883 = r725879 - r725882;
        double r725884 = r725878 / r725883;
        return r725884;
}

↓

double f(double x, double y, double z, double t) {
        double r725885 = x;
        double r725886 = -r725885;
        double r725887 = t;
        double r725888 = z;
        double r725889 = y;
        double r725890 = -r725889;
        double r725891 = fma(r725887, r725888, r725890);
        double r725892 = r725886 / r725891;
        return r725892;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	3.0
Target	2.0
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.13783064348764444 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

Initial program 3.0
\[\frac{x}{y - z \cdot t}\]
Using strategy rm
Applied frac-2neg3.0
\[\leadsto \color{blue}{\frac{-x}{-\left(y - z \cdot t\right)}}\]
Simplified2.9
\[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]
Final simplification2.9
\[\leadsto \frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2020020 +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 -1.618195973607049e+50) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t)))))

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