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

↓

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

\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 r508220 = x;
        double r508221 = y;
        double r508222 = z;
        double r508223 = t;
        double r508224 = r508222 * r508223;
        double r508225 = r508221 - r508224;
        double r508226 = r508220 / r508225;
        return r508226;
}

↓

double f(double x, double y, double z, double t) {
        double r508227 = x;
        double r508228 = -r508227;
        double r508229 = t;
        double r508230 = z;
        double r508231 = y;
        double r508232 = -r508231;
        double r508233 = fma(r508229, r508230, r508232);
        double r508234 = r508228 / r508233;
        return r508234;
}

Error

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

Target

Original	2.6
Target	1.6
Herbie	2.6

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

Derivation

Initial program 2.6
\[\frac{x}{y - z \cdot t}\]
Using strategy rm
Applied frac-2neg2.6
\[\leadsto \color{blue}{\frac{-x}{-\left(y - z \cdot t\right)}}\]
Simplified2.6
\[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]
Final simplification2.6
\[\leadsto \frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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