Average Error: 20.3 → 9.1
Time: 20.0s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2677.341884202243818435817956924438476562:\\ \;\;\;\;\left(\frac{b}{c \cdot z} - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + \frac{9 \cdot y}{\frac{z}{x} \cdot c}\\ \mathbf{elif}\;y \le 6.00607809925324836764388487972401616407 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(\frac{9}{\frac{z}{y \cdot x}} + \frac{b}{z}\right) - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;y \le 3.128028375703178244467221755223697213558 \cdot 10^{46}:\\ \;\;\;\;\left(\frac{1}{c \cdot z} \cdot b - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + 9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{elif}\;y \le 9.140692619985110186615534909303371464906 \cdot 10^{84}:\\ \;\;\;\;\frac{9}{c} \cdot \frac{x}{\frac{z}{y}} + \left(\frac{\frac{b}{c}}{z} - 4 \cdot \frac{t}{\frac{c}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + \frac{9 \cdot y}{\frac{z}{x} \cdot c}\\ \end{array}\]
\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}\;y \le -2677.341884202243818435817956924438476562:\\
\;\;\;\;\left(\frac{b}{c \cdot z} - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + \frac{9 \cdot y}{\frac{z}{x} \cdot c}\\

\mathbf{elif}\;y \le 6.00607809925324836764388487972401616407 \cdot 10^{-41}:\\
\;\;\;\;\frac{\left(\frac{9}{\frac{z}{y \cdot x}} + \frac{b}{z}\right) - \left(a \cdot 4\right) \cdot t}{c}\\

\mathbf{elif}\;y \le 3.128028375703178244467221755223697213558 \cdot 10^{46}:\\
\;\;\;\;\left(\frac{1}{c \cdot z} \cdot b - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + 9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\

\mathbf{elif}\;y \le 9.140692619985110186615534909303371464906 \cdot 10^{84}:\\
\;\;\;\;\frac{9}{c} \cdot \frac{x}{\frac{z}{y}} + \left(\frac{\frac{b}{c}}{z} - 4 \cdot \frac{t}{\frac{c}{a}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r596442 = x;
        double r596443 = 9.0;
        double r596444 = r596442 * r596443;
        double r596445 = y;
        double r596446 = r596444 * r596445;
        double r596447 = z;
        double r596448 = 4.0;
        double r596449 = r596447 * r596448;
        double r596450 = t;
        double r596451 = r596449 * r596450;
        double r596452 = a;
        double r596453 = r596451 * r596452;
        double r596454 = r596446 - r596453;
        double r596455 = b;
        double r596456 = r596454 + r596455;
        double r596457 = c;
        double r596458 = r596447 * r596457;
        double r596459 = r596456 / r596458;
        return r596459;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r596460 = y;
        double r596461 = -2677.341884202244;
        bool r596462 = r596460 <= r596461;
        double r596463 = b;
        double r596464 = c;
        double r596465 = z;
        double r596466 = r596464 * r596465;
        double r596467 = r596463 / r596466;
        double r596468 = a;
        double r596469 = 4.0;
        double r596470 = r596468 * r596469;
        double r596471 = t;
        double r596472 = r596470 * r596471;
        double r596473 = r596472 / r596464;
        double r596474 = r596467 - r596473;
        double r596475 = 9.0;
        double r596476 = r596475 * r596460;
        double r596477 = x;
        double r596478 = r596465 / r596477;
        double r596479 = r596478 * r596464;
        double r596480 = r596476 / r596479;
        double r596481 = r596474 + r596480;
        double r596482 = 6.006078099253248e-41;
        bool r596483 = r596460 <= r596482;
        double r596484 = r596460 * r596477;
        double r596485 = r596465 / r596484;
        double r596486 = r596475 / r596485;
        double r596487 = r596463 / r596465;
        double r596488 = r596486 + r596487;
        double r596489 = r596488 - r596472;
        double r596490 = r596489 / r596464;
        double r596491 = 3.128028375703178e+46;
        bool r596492 = r596460 <= r596491;
        double r596493 = 1.0;
        double r596494 = r596493 / r596466;
        double r596495 = r596494 * r596463;
        double r596496 = r596495 - r596473;
        double r596497 = r596460 / r596466;
        double r596498 = r596477 * r596497;
        double r596499 = r596475 * r596498;
        double r596500 = r596496 + r596499;
        double r596501 = 9.14069261998511e+84;
        bool r596502 = r596460 <= r596501;
        double r596503 = r596475 / r596464;
        double r596504 = r596465 / r596460;
        double r596505 = r596477 / r596504;
        double r596506 = r596503 * r596505;
        double r596507 = r596463 / r596464;
        double r596508 = r596507 / r596465;
        double r596509 = r596464 / r596468;
        double r596510 = r596471 / r596509;
        double r596511 = r596469 * r596510;
        double r596512 = r596508 - r596511;
        double r596513 = r596506 + r596512;
        double r596514 = r596502 ? r596513 : r596481;
        double r596515 = r596492 ? r596500 : r596514;
        double r596516 = r596483 ? r596490 : r596515;
        double r596517 = r596462 ? r596481 : r596516;
        return r596517;
}

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

Original20.3
Target14.2
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -2677.341884202244 or 9.14069261998511e+84 < y

    1. Initial program 26.2

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

    2. Simplified20.7

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

    4. Applied associate-*l*20.7

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

    5. Simplified20.7

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

    6. Taylor expanded around 0 18.5

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

    7. Simplified12.7

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

    8. Taylor expanded around 0 12.7

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

    9. Simplified10.5

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

    if -2677.341884202244 < y < 6.006078099253248e-41

    1. Initial program 16.5

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

    2. Simplified8.0

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

    4. Applied associate-*l*7.9

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

    5. Simplified7.9

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

    6. Taylor expanded around 0 7.9

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

    7. Simplified8.0

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

    if 6.006078099253248e-41 < y < 3.128028375703178e+46

    1. Initial program 17.4

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

    2. Simplified11.3

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

    4. Applied associate-*l*11.3

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

    5. Simplified11.3

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

    6. Taylor expanded around 0 9.4

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

    7. Simplified9.3

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

    9. Applied *-un-lft-identity9.3

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

    10. Applied times-frac9.3

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

    11. Simplified9.3

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

    12. Simplified9.3

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

    14. Applied div-inv9.5

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

    if 3.128028375703178e+46 < y < 9.14069261998511e+84

    1. Initial program 21.7

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 12.6

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

    4. Simplified10.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2677.341884202243818435817956924438476562:\\ \;\;\;\;\left(\frac{b}{c \cdot z} - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + \frac{9 \cdot y}{\frac{z}{x} \cdot c}\\ \mathbf{elif}\;y \le 6.00607809925324836764388487972401616407 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(\frac{9}{\frac{z}{y \cdot x}} + \frac{b}{z}\right) - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;y \le 3.128028375703178244467221755223697213558 \cdot 10^{46}:\\ \;\;\;\;\left(\frac{1}{c \cdot z} \cdot b - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + 9 \cdot \left(x \cdot \frac{y}{c \cdot z}\right)\\ \mathbf{elif}\;y \le 9.140692619985110186615534909303371464906 \cdot 10^{84}:\\ \;\;\;\;\frac{9}{c} \cdot \frac{x}{\frac{z}{y}} + \left(\frac{\frac{b}{c}}{z} - 4 \cdot \frac{t}{\frac{c}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} - \frac{\left(a \cdot 4\right) \cdot t}{c}\right) + \frac{9 \cdot y}{\frac{z}{x} \cdot c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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)))