Average Error: 20.2 → 8.8
Time: 21.6s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.402350092041883275668460169588838453823 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\ \mathbf{elif}\;z \le 2.285273688348913568491691630401286761171 \cdot 10^{-97}:\\ \;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\ \end{array}\]
\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}\;z \le -4.402350092041883275668460169588838453823 \cdot 10^{-72}:\\
\;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\

\mathbf{elif}\;z \le 2.285273688348913568491691630401286761171 \cdot 10^{-97}:\\
\;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}{z \cdot c}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r35494864 = x;
        double r35494865 = 9.0;
        double r35494866 = r35494864 * r35494865;
        double r35494867 = y;
        double r35494868 = r35494866 * r35494867;
        double r35494869 = z;
        double r35494870 = 4.0;
        double r35494871 = r35494869 * r35494870;
        double r35494872 = t;
        double r35494873 = r35494871 * r35494872;
        double r35494874 = a;
        double r35494875 = r35494873 * r35494874;
        double r35494876 = r35494868 - r35494875;
        double r35494877 = b;
        double r35494878 = r35494876 + r35494877;
        double r35494879 = c;
        double r35494880 = r35494869 * r35494879;
        double r35494881 = r35494878 / r35494880;
        return r35494881;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r35494882 = z;
        double r35494883 = -4.4023500920418833e-72;
        bool r35494884 = r35494882 <= r35494883;
        double r35494885 = 1.0;
        double r35494886 = r35494885 / r35494882;
        double r35494887 = b;
        double r35494888 = y;
        double r35494889 = x;
        double r35494890 = 9.0;
        double r35494891 = r35494889 * r35494890;
        double r35494892 = r35494888 * r35494891;
        double r35494893 = r35494887 + r35494892;
        double r35494894 = r35494886 * r35494893;
        double r35494895 = 4.0;
        double r35494896 = t;
        double r35494897 = r35494895 * r35494896;
        double r35494898 = a;
        double r35494899 = r35494897 * r35494898;
        double r35494900 = r35494894 - r35494899;
        double r35494901 = c;
        double r35494902 = r35494900 / r35494901;
        double r35494903 = 2.2852736883489136e-97;
        bool r35494904 = r35494882 <= r35494903;
        double r35494905 = r35494882 * r35494895;
        double r35494906 = r35494905 * r35494896;
        double r35494907 = r35494906 * r35494898;
        double r35494908 = r35494892 - r35494907;
        double r35494909 = r35494887 + r35494908;
        double r35494910 = r35494882 * r35494901;
        double r35494911 = r35494909 / r35494910;
        double r35494912 = r35494904 ? r35494911 : r35494902;
        double r35494913 = r35494884 ? r35494902 : r35494912;
        return r35494913;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target14.2
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.4023500920418833e-72 or 2.2852736883489136e-97 < z

    1. Initial program 24.8

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

    2. Simplified9.5

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

    4. Applied div-inv9.5

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

    if -4.4023500920418833e-72 < z < 2.2852736883489136e-97

    1. Initial program 6.4

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.402350092041883275668460169588838453823 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\ \mathbf{elif}\;z \le 2.285273688348913568491691630401286761171 \cdot 10^{-97}:\\ \;\;\;\;\frac{b + \left(y \cdot \left(x \cdot 9\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \left(b + y \cdot \left(x \cdot 9\right)\right) - \left(4 \cdot t\right) \cdot a}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))