Average Error: 8.0 → 8.0
Time: 11.2s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\frac{\frac{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}{a}}{2} \cdot \sqrt{1}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{\frac{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}{a}}{2} \cdot \sqrt{1}
double f(double x, double y, double z, double t, double a) {
        double r643959 = x;
        double r643960 = y;
        double r643961 = r643959 * r643960;
        double r643962 = z;
        double r643963 = 9.0;
        double r643964 = r643962 * r643963;
        double r643965 = t;
        double r643966 = r643964 * r643965;
        double r643967 = r643961 - r643966;
        double r643968 = a;
        double r643969 = 2.0;
        double r643970 = r643968 * r643969;
        double r643971 = r643967 / r643970;
        return r643971;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r643972 = x;
        double r643973 = y;
        double r643974 = z;
        double r643975 = 9.0;
        double r643976 = r643974 * r643975;
        double r643977 = t;
        double r643978 = r643976 * r643977;
        double r643979 = -r643978;
        double r643980 = fma(r643972, r643973, r643979);
        double r643981 = a;
        double r643982 = r643980 / r643981;
        double r643983 = 2.0;
        double r643984 = r643982 / r643983;
        double r643985 = 1.0;
        double r643986 = sqrt(r643985);
        double r643987 = r643984 * r643986;
        return r643987;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original8.0
Target5.9
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Initial program 8.0

    \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity8.0

    \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

  4. Applied times-frac8.1

    \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity8.1

    \[\leadsto \frac{1}{\color{blue}{1 \cdot a}} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\]

  7. Applied add-sqr-sqrt8.1

    \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\]

  8. Applied times-frac8.1

    \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{a}\right)} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\]

  9. Applied associate-*l*8.1

    \[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\right)}\]

  10. Simplified8.0

    \[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}{a}}{2}}\]

  11. Final simplification8.0

    \[\leadsto \frac{\frac{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}{a}}{2} \cdot \sqrt{1}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))