Average Error: 20.7 → 7.9
Time: 6.2s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.8565430791074941 \cdot 10^{-106} \lor \neg \left(z \le 14253.616584509742\right):\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)\\ \end{array}\]
\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}\;z \le -5.8565430791074941 \cdot 10^{-106} \lor \neg \left(z \le 14253.616584509742\right):\\
\;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r748231 = x;
        double r748232 = 9.0;
        double r748233 = r748231 * r748232;
        double r748234 = y;
        double r748235 = r748233 * r748234;
        double r748236 = z;
        double r748237 = 4.0;
        double r748238 = r748236 * r748237;
        double r748239 = t;
        double r748240 = r748238 * r748239;
        double r748241 = a;
        double r748242 = r748240 * r748241;
        double r748243 = r748235 - r748242;
        double r748244 = b;
        double r748245 = r748243 + r748244;
        double r748246 = c;
        double r748247 = r748236 * r748246;
        double r748248 = r748245 / r748247;
        return r748248;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r748249 = z;
        double r748250 = -5.856543079107494e-106;
        bool r748251 = r748249 <= r748250;
        double r748252 = 14253.616584509742;
        bool r748253 = r748249 <= r748252;
        double r748254 = !r748253;
        bool r748255 = r748251 || r748254;
        double r748256 = b;
        double r748257 = r748256 / r748249;
        double r748258 = c;
        double r748259 = r748257 / r748258;
        double r748260 = 9.0;
        double r748261 = x;
        double r748262 = y;
        double r748263 = cbrt(r748262);
        double r748264 = r748263 * r748263;
        double r748265 = r748249 / r748264;
        double r748266 = r748258 / r748263;
        double r748267 = r748265 * r748266;
        double r748268 = r748261 / r748267;
        double r748269 = r748260 * r748268;
        double r748270 = r748259 + r748269;
        double r748271 = 4.0;
        double r748272 = a;
        double r748273 = t;
        double r748274 = r748272 * r748273;
        double r748275 = r748274 / r748258;
        double r748276 = r748271 * r748275;
        double r748277 = r748270 - r748276;
        double r748278 = r748249 * r748258;
        double r748279 = r748256 / r748278;
        double r748280 = r748261 * r748262;
        double r748281 = r748280 / r748278;
        double r748282 = r748260 * r748281;
        double r748283 = r748279 + r748282;
        double r748284 = 1.0;
        double r748285 = r748284 / r748258;
        double r748286 = r748274 * r748285;
        double r748287 = r748271 * r748286;
        double r748288 = r748283 - r748287;
        double r748289 = r748255 ? r748277 : r748288;
        return r748289;
}

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

Original20.7
Target14.6
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.856543079107494e-106 or 14253.616584509742 < z

    1. Initial program 27.2

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

    2. Taylor expanded around 0 13.0

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

    4. Applied associate-/l*10.8

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

    6. Applied add-cube-cbrt11.0

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

    7. Applied times-frac9.2

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

    9. Applied associate-/r*7.3

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

    if -5.856543079107494e-106 < z < 14253.616584509742

    1. Initial program 6.0

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

    2. Taylor expanded around 0 9.2

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

    4. Applied div-inv9.2

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.8565430791074941 \cdot 10^{-106} \lor \neg \left(z \le 14253.616584509742\right):\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)) -1.1001567408041051e-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)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* 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)))