Average Error: 20.4 → 10.6
Time: 18.6s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1.447400587919281219909541782848661125831 \cdot 10^{-170}:\\ \;\;\;\;\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{c} \cdot t\\ \mathbf{elif}\;y \le 2.987715643773087253684051056069467320818 \cdot 10^{53} \lor \neg \left(y \le 1.272319050324034726610772340544411659985 \cdot 10^{244}\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \end{array}\]
\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1.447400587919281219909541782848661125831 \cdot 10^{-170}:\\
\;\;\;\;\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{c} \cdot t\\

\mathbf{elif}\;y \le 2.987715643773087253684051056069467320818 \cdot 10^{53} \lor \neg \left(y \le 1.272319050324034726610772340544411659985 \cdot 10^{244}\right):\\
\;\;\;\;\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r397782 = x;
        double r397783 = 9.0;
        double r397784 = r397782 * r397783;
        double r397785 = y;
        double r397786 = r397784 * r397785;
        double r397787 = z;
        double r397788 = 4.0;
        double r397789 = r397787 * r397788;
        double r397790 = t;
        double r397791 = r397789 * r397790;
        double r397792 = a;
        double r397793 = r397791 * r397792;
        double r397794 = r397786 - r397793;
        double r397795 = b;
        double r397796 = r397794 + r397795;
        double r397797 = c;
        double r397798 = r397787 * r397797;
        double r397799 = r397796 / r397798;
        return r397799;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r397800 = y;
        double r397801 = 1.4474005879192812e-170;
        bool r397802 = r397800 <= r397801;
        double r397803 = 9.0;
        double r397804 = x;
        double r397805 = r397804 * r397800;
        double r397806 = z;
        double r397807 = c;
        double r397808 = r397806 * r397807;
        double r397809 = r397805 / r397808;
        double r397810 = r397803 * r397809;
        double r397811 = b;
        double r397812 = r397811 / r397808;
        double r397813 = r397810 + r397812;
        double r397814 = a;
        double r397815 = 4.0;
        double r397816 = r397814 * r397815;
        double r397817 = r397816 / r397807;
        double r397818 = t;
        double r397819 = r397817 * r397818;
        double r397820 = r397813 - r397819;
        double r397821 = 2.9877156437730873e+53;
        bool r397822 = r397800 <= r397821;
        double r397823 = 1.2723190503240347e+244;
        bool r397824 = r397800 <= r397823;
        double r397825 = !r397824;
        bool r397826 = r397822 || r397825;
        double r397827 = r397804 * r397803;
        double r397828 = fma(r397800, r397827, r397811);
        double r397829 = r397828 / r397806;
        double r397830 = r397816 * r397818;
        double r397831 = r397829 - r397830;
        double r397832 = 1.0;
        double r397833 = r397832 / r397807;
        double r397834 = r397831 * r397833;
        double r397835 = r397804 / r397806;
        double r397836 = r397800 / r397807;
        double r397837 = r397835 * r397836;
        double r397838 = fma(r397837, r397803, r397812);
        double r397839 = r397807 / r397818;
        double r397840 = r397816 / r397839;
        double r397841 = r397838 - r397840;
        double r397842 = r397826 ? r397834 : r397841;
        double r397843 = r397802 ? r397820 : r397842;
        return r397843;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original20.4
Target14.4
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < 1.4474005879192812e-170

    1. Initial program 19.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified12.4

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]

    3. Taylor expanded around 0 10.8

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]

    4. Simplified10.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}}\]
    5. Using strategy rm

    6. Applied associate-/l*9.9

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \color{blue}{\frac{a \cdot 4}{\frac{c}{t}}}\]
    7. Using strategy rm

    8. Applied fma-udef9.9

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z \cdot c} \cdot 9 + \frac{b}{z \cdot c}\right)} - \frac{a \cdot 4}{\frac{c}{t}}\]

    9. Simplified9.9

      \[\leadsto \left(\color{blue}{9 \cdot \frac{x \cdot y}{z \cdot c}} + \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\]
    10. Using strategy rm

    11. Applied associate-/r/10.2

      \[\leadsto \left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - \color{blue}{\frac{a \cdot 4}{c} \cdot t}\]

    if 1.4474005879192812e-170 < y < 2.9877156437730873e+53 or 1.2723190503240347e+244 < y

    1. Initial program 19.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified11.5

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied div-inv11.6

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}}\]

    if 2.9877156437730873e+53 < y < 1.2723190503240347e+244

    1. Initial program 26.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified20.5

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]

    3. Taylor expanded around 0 18.4

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]

    4. Simplified18.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}}\]
    5. Using strategy rm

    6. Applied associate-/l*17.8

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \color{blue}{\frac{a \cdot 4}{\frac{c}{t}}}\]
    7. Using strategy rm

    8. Applied times-frac10.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{z} \cdot \frac{y}{c}}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.447400587919281219909541782848661125831 \cdot 10^{-170}:\\ \;\;\;\;\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{c} \cdot t\\ \mathbf{elif}\;y \le 2.987715643773087253684051056069467320818 \cdot 10^{53} \lor \neg \left(y \le 1.272319050324034726610772340544411659985 \cdot 10^{244}\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041049e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))