Average Error: 20.6 → 9.1
Time: 5.3s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -7.78888637189634463 \cdot 10^{180}:\\ \;\;\;\;\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) + \left(\frac{b}{z \cdot c} - 4 \cdot \frac{a \cdot t}{c}\right)\\ \mathbf{elif}\;c \le -1.61911827812599734 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \mathbf{elif}\;c \le 3.67640605479528566 \cdot 10^{28}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c} + \left(\frac{b}{z \cdot c} - 4 \cdot \left(a \cdot \frac{t}{c}\right)\right)\\ \end{array}\]
\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}\;c \le -7.78888637189634463 \cdot 10^{180}:\\
\;\;\;\;\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) + \left(\frac{b}{z \cdot c} - 4 \cdot \frac{a \cdot t}{c}\right)\\

\mathbf{elif}\;c \le -1.61911827812599734 \cdot 10^{-22}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\

\mathbf{elif}\;c \le 3.67640605479528566 \cdot 10^{28}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r843241 = x;
        double r843242 = 9.0;
        double r843243 = r843241 * r843242;
        double r843244 = y;
        double r843245 = r843243 * r843244;
        double r843246 = z;
        double r843247 = 4.0;
        double r843248 = r843246 * r843247;
        double r843249 = t;
        double r843250 = r843248 * r843249;
        double r843251 = a;
        double r843252 = r843250 * r843251;
        double r843253 = r843245 - r843252;
        double r843254 = b;
        double r843255 = r843253 + r843254;
        double r843256 = c;
        double r843257 = r843246 * r843256;
        double r843258 = r843255 / r843257;
        return r843258;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r843259 = c;
        double r843260 = -7.788886371896345e+180;
        bool r843261 = r843259 <= r843260;
        double r843262 = 9.0;
        double r843263 = cbrt(r843262);
        double r843264 = r843263 * r843263;
        double r843265 = x;
        double r843266 = z;
        double r843267 = r843265 / r843266;
        double r843268 = y;
        double r843269 = r843268 / r843259;
        double r843270 = r843267 * r843269;
        double r843271 = r843263 * r843270;
        double r843272 = r843264 * r843271;
        double r843273 = b;
        double r843274 = r843266 * r843259;
        double r843275 = r843273 / r843274;
        double r843276 = 4.0;
        double r843277 = a;
        double r843278 = t;
        double r843279 = r843277 * r843278;
        double r843280 = r843279 / r843259;
        double r843281 = r843276 * r843280;
        double r843282 = r843275 - r843281;
        double r843283 = r843272 + r843282;
        double r843284 = -1.6191182781259973e-22;
        bool r843285 = r843259 <= r843284;
        double r843286 = -r843276;
        double r843287 = r843259 / r843277;
        double r843288 = r843278 / r843287;
        double r843289 = r843262 * r843268;
        double r843290 = fma(r843265, r843289, r843273);
        double r843291 = r843290 / r843274;
        double r843292 = fma(r843286, r843288, r843291);
        double r843293 = 3.6764060547952857e+28;
        bool r843294 = r843259 <= r843293;
        double r843295 = r843278 * r843277;
        double r843296 = r843295 / r843259;
        double r843297 = r843262 * r843265;
        double r843298 = fma(r843297, r843268, r843273);
        double r843299 = r843298 / r843266;
        double r843300 = r843299 / r843259;
        double r843301 = fma(r843286, r843296, r843300);
        double r843302 = r843265 * r843268;
        double r843303 = r843302 / r843274;
        double r843304 = r843262 * r843303;
        double r843305 = r843278 / r843259;
        double r843306 = r843277 * r843305;
        double r843307 = r843276 * r843306;
        double r843308 = r843275 - r843307;
        double r843309 = r843304 + r843308;
        double r843310 = r843294 ? r843301 : r843309;
        double r843311 = r843285 ? r843292 : r843310;
        double r843312 = r843261 ? r843283 : r843311;
        return r843312;
}

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

Original20.6
Target14.6
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if c < -7.788886371896345e+180

    1. Initial program 26.5

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

    2. Simplified18.6

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

    3. Taylor expanded around 0 18.5

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

    4. Simplified18.5

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

    6. Applied fma-udef18.5

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

    8. Applied add-cube-cbrt18.5

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

    9. Applied associate-*l*18.5

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

    11. Applied times-frac18.4

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

    if -7.788886371896345e+180 < c < -1.6191182781259973e-22

    1. Initial program 19.6

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

    2. Simplified10.8

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

    4. Applied associate-/l*8.3

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

    if -1.6191182781259973e-22 < c < 3.6764060547952857e+28

    1. Initial program 14.9

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

    2. Simplified5.2

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

    4. Applied associate-/r*2.7

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

    5. Simplified2.7

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

    if 3.6764060547952857e+28 < c

    1. Initial program 24.3

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

    2. Simplified15.6

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

    3. Taylor expanded around 0 15.5

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

    4. Simplified15.5

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

    6. Applied fma-udef15.5

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

    8. Applied *-un-lft-identity15.5

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

    9. Applied times-frac11.5

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

    10. Simplified11.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -7.78888637189634463 \cdot 10^{180}:\\ \;\;\;\;\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) + \left(\frac{b}{z \cdot c} - 4 \cdot \frac{a \cdot t}{c}\right)\\ \mathbf{elif}\;c \le -1.61911827812599734 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \mathbf{elif}\;c \le 3.67640605479528566 \cdot 10^{28}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c} + \left(\frac{b}{z \cdot c} - 4 \cdot \left(a \cdot \frac{t}{c}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)) -1.1001567408041051e-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)))