Average Error: 20.8 → 10.8
Time: 5.2s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.36732518833848025 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 7.9555437739864164 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.24947951631066874 \cdot 10^{61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]
\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 -6.36732518833848025 \cdot 10^{-192}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\
\;\;\;\;\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 7.9555437739864164 \cdot 10^{-71}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;t \le 2.24947951631066874 \cdot 10^{61}:\\
\;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r808507 = x;
        double r808508 = 9.0;
        double r808509 = r808507 * r808508;
        double r808510 = y;
        double r808511 = r808509 * r808510;
        double r808512 = z;
        double r808513 = 4.0;
        double r808514 = r808512 * r808513;
        double r808515 = t;
        double r808516 = r808514 * r808515;
        double r808517 = a;
        double r808518 = r808516 * r808517;
        double r808519 = r808511 - r808518;
        double r808520 = b;
        double r808521 = r808519 + r808520;
        double r808522 = c;
        double r808523 = r808512 * r808522;
        double r808524 = r808521 / r808523;
        return r808524;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r808525 = t;
        double r808526 = -6.36732518833848e-192;
        bool r808527 = r808525 <= r808526;
        double r808528 = b;
        double r808529 = z;
        double r808530 = c;
        double r808531 = r808529 * r808530;
        double r808532 = r808528 / r808531;
        double r808533 = 9.0;
        double r808534 = 1.0;
        double r808535 = r808534 / r808529;
        double r808536 = x;
        double r808537 = y;
        double r808538 = r808530 / r808537;
        double r808539 = r808536 / r808538;
        double r808540 = r808535 * r808539;
        double r808541 = r808533 * r808540;
        double r808542 = r808532 + r808541;
        double r808543 = 4.0;
        double r808544 = a;
        double r808545 = r808525 / r808530;
        double r808546 = r808544 * r808545;
        double r808547 = r808543 * r808546;
        double r808548 = r808542 - r808547;
        double r808549 = 3.507527873864621e-285;
        bool r808550 = r808525 <= r808549;
        double r808551 = r808536 * r808533;
        double r808552 = r808551 * r808537;
        double r808553 = r808529 * r808543;
        double r808554 = r808553 * r808525;
        double r808555 = r808554 * r808544;
        double r808556 = r808552 - r808555;
        double r808557 = r808556 + r808528;
        double r808558 = r808557 / r808529;
        double r808559 = r808558 / r808530;
        double r808560 = 7.955543773986416e-71;
        bool r808561 = r808525 <= r808560;
        double r808562 = r808536 / r808531;
        double r808563 = r808562 * r808537;
        double r808564 = r808533 * r808563;
        double r808565 = r808532 + r808564;
        double r808566 = r808565 - r808547;
        double r808567 = 2.2494795163106687e+61;
        bool r808568 = r808525 <= r808567;
        double r808569 = cbrt(r808528);
        double r808570 = r808569 * r808569;
        double r808571 = r808570 / r808529;
        double r808572 = r808569 / r808530;
        double r808573 = r808571 * r808572;
        double r808574 = r808536 * r808537;
        double r808575 = r808574 / r808531;
        double r808576 = r808533 * r808575;
        double r808577 = r808573 + r808576;
        double r808578 = r808544 * r808525;
        double r808579 = r808578 / r808530;
        double r808580 = r808543 * r808579;
        double r808581 = r808577 - r808580;
        double r808582 = r808568 ? r808581 : r808548;
        double r808583 = r808561 ? r808566 : r808582;
        double r808584 = r808550 ? r808559 : r808583;
        double r808585 = r808527 ? r808548 : r808584;
        return r808585;
}

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

Original20.8
Target14.9
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -6.36732518833848e-192 or 2.2494795163106687e+61 < t

    1. Initial program 25.4

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

    2. Taylor expanded around 0 14.0

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

    4. Applied *-un-lft-identity14.0

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

    5. Applied times-frac11.7

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

    6. Simplified11.7

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

    8. Applied associate-/l*9.7

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

    10. Applied *-un-lft-identity9.7

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

    11. Applied times-frac10.0

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

    12. Applied *-un-lft-identity10.0

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

    13. Applied times-frac10.6

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

    14. Simplified10.6

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

    if -6.36732518833848e-192 < t < 3.507527873864621e-285

    1. Initial program 12.6

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

    3. Applied associate-/r*10.9

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

    if 3.507527873864621e-285 < t < 7.955543773986416e-71

    1. Initial program 13.7

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

    2. Taylor expanded around 0 10.1

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

    4. Applied *-un-lft-identity10.1

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

    5. Applied times-frac12.2

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

    6. Simplified12.2

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

    8. Applied associate-/l*11.4

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

    10. Applied associate-/r/12.2

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

    if 7.955543773986416e-71 < t < 2.2494795163106687e+61

    1. Initial program 18.8

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

    2. Taylor expanded around 0 9.6

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

    4. Applied add-cube-cbrt9.9

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

    5. Applied times-frac9.5

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.36732518833848025 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 7.9555437739864164 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.24947951631066874 \cdot 10^{61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))