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

↓

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

\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 r723997 = x;
        double r723998 = y;
        double r723999 = z;
        double r724000 = t;
        double r724001 = r723999 * r724000;
        double r724002 = r723998 - r724001;
        double r724003 = r723997 / r724002;
        return r724003;
}

↓

double f(double x, double y, double z, double t) {
        double r724004 = x;
        double r724005 = -r724004;
        double r724006 = t;
        double r724007 = z;
        double r724008 = y;
        double r724009 = -r724008;
        double r724010 = fma(r724006, r724007, r724009);
        double r724011 = r724005 / r724010;
        return r724011;
}

Error

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

Target

Original	2.9
Target	1.9
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 2.9
\[\frac{x}{y - z \cdot t}\]
Using strategy rm
Applied frac-2neg2.9
\[\leadsto \color{blue}{\frac{-x}{-\left(y - z \cdot t\right)}}\]
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 2020027 +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))))