Average Error: 2.9 → 3.0
Time: 2.9s
Precision: 64
\[\frac{x}{y - z \cdot t}\]
↓
\[\left(-x\right) \cdot \frac{1}{\mathsf{fma}\left(t, z, -y\right)}\]
\frac{x}{y - z \cdot t}
↓
\left(-x\right) \cdot \frac{1}{\mathsf{fma}\left(t, z, -y\right)}
double f(double x, double y, double z, double t) {
        double r814162 = x;
        double r814163 = y;
        double r814164 = z;
        double r814165 = t;
        double r814166 = r814164 * r814165;
        double r814167 = r814163 - r814166;
        double r814168 = r814162 / r814167;
        return r814168;
}
↓
double f(double x, double y, double z, double t) {
        double r814169 = x;
        double r814170 = -r814169;
        double r814171 = 1.0;
        double r814172 = t;
        double r814173 = z;
        double r814174 = y;
        double r814175 = -r814174;
        double r814176 = fma(r814172, r814173, r814175);
        double r814177 = r814171 / r814176;
        double r814178 = r814170 * r814177;
        return r814178;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.9
Target1.7
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.13783064348764444 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

  1. Initial program 2.9

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

  3. Applied frac-2neg2.9

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

  4. Simplified2.9

    \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]
  5. Using strategy rm

  6. Applied div-inv3.0

    \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\mathsf{fma}\left(t, z, -y\right)}}\]

  7. Final simplification3.0

    \[\leadsto \left(-x\right) \cdot \frac{1}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2020057 +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))))