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

↓

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

\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 r699829 = x;
        double r699830 = y;
        double r699831 = z;
        double r699832 = t;
        double r699833 = r699831 * r699832;
        double r699834 = r699830 - r699833;
        double r699835 = r699829 / r699834;
        return r699835;
}

↓

double f(double x, double y, double z, double t) {
        double r699836 = x;
        double r699837 = -r699836;
        double r699838 = t;
        double r699839 = z;
        double r699840 = y;
        double r699841 = -r699840;
        double r699842 = fma(r699838, r699839, r699841);
        double r699843 = r699837 / r699842;
        return r699843;
}

Error

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

Target

Original	2.8
Target	1.8
Herbie	2.8

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

Derivation

Initial program 2.8
\[\frac{x}{y - z \cdot t}\]
Using strategy rm
Applied frac-2neg2.8
\[\leadsto \color{blue}{\frac{-x}{-\left(y - z \cdot t\right)}}\]
Simplified2.8
\[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]
Final simplification2.8
\[\leadsto \frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2020089 +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))))