Average Error: 21.4 → 8.6
Time: 7.5s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.979804775080449006989440510883710580261 \cdot 10^{-100}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\ \mathbf{elif}\;t \le -5.217063799066551531891142327944064129209 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 5492895832084550763595956224:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\ \end{array}\]
\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}\;t \le -1.979804775080449006989440510883710580261 \cdot 10^{-100}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\

\mathbf{elif}\;t \le -5.217063799066551531891142327944064129209 \cdot 10^{-249}:\\
\;\;\;\;\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}\;t \le 5492895832084550763595956224:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r722804 = x;
        double r722805 = 9.0;
        double r722806 = r722804 * r722805;
        double r722807 = y;
        double r722808 = r722806 * r722807;
        double r722809 = z;
        double r722810 = 4.0;
        double r722811 = r722809 * r722810;
        double r722812 = t;
        double r722813 = r722811 * r722812;
        double r722814 = a;
        double r722815 = r722813 * r722814;
        double r722816 = r722808 - r722815;
        double r722817 = b;
        double r722818 = r722816 + r722817;
        double r722819 = c;
        double r722820 = r722809 * r722819;
        double r722821 = r722818 / r722820;
        return r722821;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r722822 = t;
        double r722823 = -1.979804775080449e-100;
        bool r722824 = r722822 <= r722823;
        double r722825 = b;
        double r722826 = z;
        double r722827 = c;
        double r722828 = r722826 * r722827;
        double r722829 = r722825 / r722828;
        double r722830 = 9.0;
        double r722831 = x;
        double r722832 = y;
        double r722833 = r722828 / r722832;
        double r722834 = r722831 / r722833;
        double r722835 = r722830 * r722834;
        double r722836 = r722829 + r722835;
        double r722837 = 4.0;
        double r722838 = a;
        double r722839 = cbrt(r722827);
        double r722840 = r722839 * r722839;
        double r722841 = r722838 / r722840;
        double r722842 = cbrt(r722822);
        double r722843 = r722842 * r722842;
        double r722844 = cbrt(r722840);
        double r722845 = r722843 / r722844;
        double r722846 = r722841 * r722845;
        double r722847 = cbrt(r722839);
        double r722848 = r722842 / r722847;
        double r722849 = r722846 * r722848;
        double r722850 = r722837 * r722849;
        double r722851 = r722836 - r722850;
        double r722852 = -5.217063799066552e-249;
        bool r722853 = r722822 <= r722852;
        double r722854 = r722831 * r722830;
        double r722855 = r722854 * r722832;
        double r722856 = r722826 * r722837;
        double r722857 = r722856 * r722822;
        double r722858 = r722857 * r722838;
        double r722859 = r722855 - r722858;
        double r722860 = r722859 + r722825;
        double r722861 = r722860 / r722826;
        double r722862 = r722861 / r722827;
        double r722863 = 5.492895832084551e+27;
        bool r722864 = r722822 <= r722863;
        double r722865 = 1.0;
        double r722866 = r722833 / r722831;
        double r722867 = r722865 / r722866;
        double r722868 = r722830 * r722867;
        double r722869 = r722829 + r722868;
        double r722870 = r722838 * r722822;
        double r722871 = r722870 / r722827;
        double r722872 = r722837 * r722871;
        double r722873 = r722869 - r722872;
        double r722874 = r722864 ? r722873 : r722851;
        double r722875 = r722853 ? r722862 : r722874;
        double r722876 = r722824 ? r722851 : r722875;
        return r722876;
}

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

Original21.4
Target14.7
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.979804775080449e-100 or 5.492895832084551e+27 < t

    1. Initial program 28.0

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

    2. Taylor expanded around 0 14.1

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

    4. Applied associate-/l*13.1

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

    6. Applied add-cube-cbrt13.5

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

    7. Applied times-frac8.1

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

    9. Applied add-cube-cbrt8.1

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

    10. Applied cbrt-prod8.2

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

    11. Applied add-cube-cbrt8.3

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

    12. Applied times-frac8.2

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

    13. Applied associate-*r*7.4

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

    if -1.979804775080449e-100 < t < -5.217063799066552e-249

    1. Initial program 14.4

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

    3. Applied associate-/r*11.7

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

    if -5.217063799066552e-249 < t < 5.492895832084551e+27

    1. Initial program 14.3

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

    2. Taylor expanded around 0 9.8

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

    4. Applied associate-/l*9.1

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

    6. Applied clear-num9.2

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{1}{\frac{\frac{z \cdot c}{y}}{x}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.979804775080449006989440510883710580261 \cdot 10^{-100}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\ \mathbf{elif}\;t \le -5.217063799066551531891142327944064129209 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 5492895832084550763595956224:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{1}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c} \cdot \sqrt[3]{c}}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{c}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))