Average Error: 20.7 → 4.8
Time: 6.5s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9, x \cdot \frac{y}{z}, \frac{b}{z}\right)}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -6.6758011168071027 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.9175736342481858 \cdot 10^{-225}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.4854174770591123 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9, x \cdot \frac{y}{z}, \frac{b}{z}\right)}{c}\right)\\ \end{array}\]
\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}\;\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} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9, x \cdot \frac{y}{z}, \frac{b}{z}\right)}{c}\right)\\

\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} \le -6.6758011168071027 \cdot 10^{-31}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\

\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} \le 2.9175736342481858 \cdot 10^{-225}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right)\\

\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} \le 1.4854174770591123 \cdot 10^{307}:\\
\;\;\;\;\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}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r688231 = x;
        double r688232 = 9.0;
        double r688233 = r688231 * r688232;
        double r688234 = y;
        double r688235 = r688233 * r688234;
        double r688236 = z;
        double r688237 = 4.0;
        double r688238 = r688236 * r688237;
        double r688239 = t;
        double r688240 = r688238 * r688239;
        double r688241 = a;
        double r688242 = r688240 * r688241;
        double r688243 = r688235 - r688242;
        double r688244 = b;
        double r688245 = r688243 + r688244;
        double r688246 = c;
        double r688247 = r688236 * r688246;
        double r688248 = r688245 / r688247;
        return r688248;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r688249 = x;
        double r688250 = 9.0;
        double r688251 = r688249 * r688250;
        double r688252 = y;
        double r688253 = r688251 * r688252;
        double r688254 = z;
        double r688255 = 4.0;
        double r688256 = r688254 * r688255;
        double r688257 = t;
        double r688258 = r688256 * r688257;
        double r688259 = a;
        double r688260 = r688258 * r688259;
        double r688261 = r688253 - r688260;
        double r688262 = b;
        double r688263 = r688261 + r688262;
        double r688264 = c;
        double r688265 = r688254 * r688264;
        double r688266 = r688263 / r688265;
        double r688267 = -inf.0;
        bool r688268 = r688266 <= r688267;
        double r688269 = -r688255;
        double r688270 = r688259 / r688264;
        double r688271 = r688257 * r688270;
        double r688272 = r688252 / r688254;
        double r688273 = r688249 * r688272;
        double r688274 = r688262 / r688254;
        double r688275 = fma(r688250, r688273, r688274);
        double r688276 = r688275 / r688264;
        double r688277 = fma(r688269, r688271, r688276);
        double r688278 = -6.675801116807103e-31;
        bool r688279 = r688266 <= r688278;
        double r688280 = r688257 * r688259;
        double r688281 = r688280 / r688264;
        double r688282 = r688249 * r688252;
        double r688283 = r688282 / r688265;
        double r688284 = r688262 / r688265;
        double r688285 = fma(r688250, r688283, r688284);
        double r688286 = fma(r688269, r688281, r688285);
        double r688287 = 2.917573634248186e-225;
        bool r688288 = r688266 <= r688287;
        double r688289 = r688264 / r688259;
        double r688290 = r688257 / r688289;
        double r688291 = r688282 / r688254;
        double r688292 = fma(r688250, r688291, r688274);
        double r688293 = r688292 / r688264;
        double r688294 = fma(r688269, r688290, r688293);
        double r688295 = 1.4854174770591123e+307;
        bool r688296 = r688266 <= r688295;
        double r688297 = r688296 ? r688266 : r688277;
        double r688298 = r688288 ? r688294 : r688297;
        double r688299 = r688279 ? r688286 : r688298;
        double r688300 = r688268 ? r688277 : r688299;
        return r688300;
}

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

Original20.7
Target14.6
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 1.4854174770591123e+307 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

    1. Initial program 63.8

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

    2. Simplified30.9

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

    4. Applied associate-/r*26.9

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

    5. Simplified27.2

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

    6. Taylor expanded around 0 26.8

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

    7. Simplified26.8

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

    9. Applied *-un-lft-identity26.8

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

    10. Applied times-frac17.7

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

    11. Simplified17.7

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

    13. Applied *-un-lft-identity17.7

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

    14. Applied times-frac13.0

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

    15. Simplified13.0

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

    if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -6.675801116807103e-31

    1. Initial program 0.5

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

    2. Simplified2.8

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

    3. Taylor expanded around 0 2.8

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

    4. Simplified2.8

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

    if -6.675801116807103e-31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.917573634248186e-225

    1. Initial program 20.5

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

    2. Simplified12.5

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

    4. Applied associate-/r*0.9

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

    5. Simplified1.0

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

    6. Taylor expanded around 0 1.0

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

    7. Simplified0.9

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

    9. Applied associate-/l*2.8

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

    if 2.917573634248186e-225 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.4854174770591123e+307

    1. Initial program 0.6

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9, x \cdot \frac{y}{z}, \frac{b}{z}\right)}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -6.6758011168071027 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.9175736342481858 \cdot 10^{-225}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t}{\frac{c}{a}}, \frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.4854174770591123 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9, x \cdot \frac{y}{z}, \frac{b}{z}\right)}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))