Average Error: 20.5 → 8.8
Time: 14.3s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.446275181309231934819127327156697311503 \cdot 10^{191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.351963005752456200525318697249689454739 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\frac{c}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.10392962055064688946472719929111178911 \cdot 10^{301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]
\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}\;\left(x \cdot 9\right) \cdot y \le -1.446275181309231934819127327156697311503 \cdot 10^{191}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 0.0:\\
\;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.351963005752456200525318697249689454739 \cdot 10^{-50}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\frac{c}{\sqrt[3]{t}}}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.10392962055064688946472719929111178911 \cdot 10^{301}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r620408 = x;
        double r620409 = 9.0;
        double r620410 = r620408 * r620409;
        double r620411 = y;
        double r620412 = r620410 * r620411;
        double r620413 = z;
        double r620414 = 4.0;
        double r620415 = r620413 * r620414;
        double r620416 = t;
        double r620417 = r620415 * r620416;
        double r620418 = a;
        double r620419 = r620417 * r620418;
        double r620420 = r620412 - r620419;
        double r620421 = b;
        double r620422 = r620420 + r620421;
        double r620423 = c;
        double r620424 = r620413 * r620423;
        double r620425 = r620422 / r620424;
        return r620425;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r620426 = x;
        double r620427 = 9.0;
        double r620428 = r620426 * r620427;
        double r620429 = y;
        double r620430 = r620428 * r620429;
        double r620431 = -1.446275181309232e+191;
        bool r620432 = r620430 <= r620431;
        double r620433 = z;
        double r620434 = r620426 / r620433;
        double r620435 = c;
        double r620436 = r620429 / r620435;
        double r620437 = r620434 * r620436;
        double r620438 = b;
        double r620439 = r620433 * r620435;
        double r620440 = r620438 / r620439;
        double r620441 = fma(r620437, r620427, r620440);
        double r620442 = a;
        double r620443 = 4.0;
        double r620444 = r620442 * r620443;
        double r620445 = t;
        double r620446 = r620444 * r620445;
        double r620447 = r620446 / r620435;
        double r620448 = r620441 - r620447;
        double r620449 = 0.0;
        bool r620450 = r620430 <= r620449;
        double r620451 = r620426 * r620429;
        double r620452 = r620451 / r620439;
        double r620453 = 1.0;
        double r620454 = r620453 / r620433;
        double r620455 = r620438 / r620435;
        double r620456 = r620454 * r620455;
        double r620457 = fma(r620452, r620427, r620456);
        double r620458 = r620435 / r620445;
        double r620459 = r620444 / r620458;
        double r620460 = r620457 - r620459;
        double r620461 = 1.3519630057524562e-50;
        bool r620462 = r620430 <= r620461;
        double r620463 = fma(r620452, r620427, r620440);
        double r620464 = cbrt(r620445);
        double r620465 = r620464 * r620464;
        double r620466 = r620444 * r620465;
        double r620467 = r620435 / r620464;
        double r620468 = r620466 / r620467;
        double r620469 = r620463 - r620468;
        double r620470 = 1.103929620550647e+301;
        bool r620471 = r620430 <= r620470;
        double r620472 = r620439 / r620429;
        double r620473 = r620426 / r620472;
        double r620474 = fma(r620473, r620427, r620440);
        double r620475 = r620474 - r620447;
        double r620476 = r620471 ? r620460 : r620475;
        double r620477 = r620462 ? r620469 : r620476;
        double r620478 = r620450 ? r620460 : r620477;
        double r620479 = r620432 ? r620448 : r620478;
        return r620479;
}

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

Original20.5
Target14.6
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804105117061698089246936481893 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (* x 9.0) y) < -1.446275181309232e+191

    1. Initial program 40.2

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

    2. Simplified37.0

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

    3. Taylor expanded around 0 33.3

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

    4. Simplified33.5

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

    6. Applied times-frac14.8

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

    if -1.446275181309232e+191 < (* (* x 9.0) y) < 0.0 or 1.3519630057524562e-50 < (* (* x 9.0) y) < 1.103929620550647e+301

    1. Initial program 17.0

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

    2. Simplified9.7

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

    3. Taylor expanded around 0 8.0

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

    4. Simplified8.1

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

    6. Applied associate-/l*7.1

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

    8. Applied *-un-lft-identity7.1

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

    9. Applied times-frac7.8

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

    if 0.0 < (* (* x 9.0) y) < 1.3519630057524562e-50

    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. Simplified7.4

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

    3. Taylor expanded around 0 7.8

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

    4. Simplified7.8

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

    6. Applied associate-/l*7.5

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

    8. Applied add-cube-cbrt8.0

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

    9. Applied *-un-lft-identity8.0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{\color{blue}{1 \cdot c}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

    10. Applied times-frac8.0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\color{blue}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{c}{\sqrt[3]{t}}}}\]

    11. Applied associate-/r*7.3

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \color{blue}{\frac{\frac{a \cdot 4}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}{\frac{c}{\sqrt[3]{t}}}}\]

    12. Simplified7.3

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

    if 1.103929620550647e+301 < (* (* x 9.0) y)

    1. Initial program 60.9

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

    2. Simplified60.7

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

    3. Taylor expanded around 0 59.4

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

    4. Simplified59.4

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

    6. Applied associate-/l*22.0

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.446275181309231934819127327156697311503 \cdot 10^{191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.351963005752456200525318697249689454739 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\frac{c}{\sqrt[3]{t}}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.10392962055064688946472719929111178911 \cdot 10^{301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{1}{z} \cdot \frac{b}{c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))