Average Error: 19.6 → 7.2
Time: 22.8s
Precision: 64
\[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -3.4143397402486294 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - 4.0 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.4789431977947666 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \mathsf{fma}\left(9.0, y \cdot x, b\right) - \left(t \cdot a\right) \cdot 4.0}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \left(a \cdot \frac{t}{c}\right) \cdot 4.0\\ \end{array}\]
\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -3.4143397402486294 \cdot 10^{-292}:\\
\;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - 4.0 \cdot \frac{a}{\frac{c}{t}}\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.4789431977947666 \cdot 10^{+196}:\\
\;\;\;\;\frac{\frac{1}{z} \cdot \mathsf{fma}\left(9.0, y \cdot x, b\right) - \left(t \cdot a\right) \cdot 4.0}{c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \left(a \cdot \frac{t}{c}\right) \cdot 4.0\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r32864494 = x;
        double r32864495 = 9.0;
        double r32864496 = r32864494 * r32864495;
        double r32864497 = y;
        double r32864498 = r32864496 * r32864497;
        double r32864499 = z;
        double r32864500 = 4.0;
        double r32864501 = r32864499 * r32864500;
        double r32864502 = t;
        double r32864503 = r32864501 * r32864502;
        double r32864504 = a;
        double r32864505 = r32864503 * r32864504;
        double r32864506 = r32864498 - r32864505;
        double r32864507 = b;
        double r32864508 = r32864506 + r32864507;
        double r32864509 = c;
        double r32864510 = r32864499 * r32864509;
        double r32864511 = r32864508 / r32864510;
        return r32864511;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r32864512 = x;
        double r32864513 = 9.0;
        double r32864514 = r32864512 * r32864513;
        double r32864515 = y;
        double r32864516 = r32864514 * r32864515;
        double r32864517 = z;
        double r32864518 = 4.0;
        double r32864519 = r32864517 * r32864518;
        double r32864520 = t;
        double r32864521 = r32864519 * r32864520;
        double r32864522 = a;
        double r32864523 = r32864521 * r32864522;
        double r32864524 = r32864516 - r32864523;
        double r32864525 = b;
        double r32864526 = r32864524 + r32864525;
        double r32864527 = c;
        double r32864528 = r32864527 * r32864517;
        double r32864529 = r32864526 / r32864528;
        double r32864530 = -3.4143397402486294e-292;
        bool r32864531 = r32864529 <= r32864530;
        double r32864532 = r32864525 / r32864528;
        double r32864533 = r32864528 / r32864515;
        double r32864534 = r32864512 / r32864533;
        double r32864535 = r32864534 * r32864513;
        double r32864536 = r32864532 + r32864535;
        double r32864537 = r32864527 / r32864520;
        double r32864538 = r32864522 / r32864537;
        double r32864539 = r32864518 * r32864538;
        double r32864540 = r32864536 - r32864539;
        double r32864541 = 1.4789431977947666e+196;
        bool r32864542 = r32864529 <= r32864541;
        double r32864543 = 1.0;
        double r32864544 = r32864543 / r32864517;
        double r32864545 = r32864515 * r32864512;
        double r32864546 = fma(r32864513, r32864545, r32864525);
        double r32864547 = r32864544 * r32864546;
        double r32864548 = r32864520 * r32864522;
        double r32864549 = r32864548 * r32864518;
        double r32864550 = r32864547 - r32864549;
        double r32864551 = r32864550 / r32864527;
        double r32864552 = r32864520 / r32864527;
        double r32864553 = r32864522 * r32864552;
        double r32864554 = r32864553 * r32864518;
        double r32864555 = r32864536 - r32864554;
        double r32864556 = r32864542 ? r32864551 : r32864555;
        double r32864557 = r32864531 ? r32864540 : r32864556;
        return r32864557;
}

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

Original19.6
Target13.7
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9.0 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9.0 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -3.4143397402486294e-292

    1. Initial program 11.7

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified11.9

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(9.0, y \cdot x, b\right)}{z} - \left(t \cdot a\right) \cdot 4.0}{c}}\]

    3. Taylor expanded around 0 6.8

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{a \cdot t}{c}}\]
    4. Using strategy rm

    5. Applied associate-/l*6.9

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{a \cdot t}{c}\]
    6. Using strategy rm

    7. Applied associate-/l*6.4

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\frac{a}{\frac{c}{t}}}\]

    if -3.4143397402486294e-292 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.4789431977947666e+196

    1. Initial program 11.3

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified4.6

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(9.0, y \cdot x, b\right)}{z} - \left(t \cdot a\right) \cdot 4.0}{c}}\]
    3. Using strategy rm

    4. Applied div-inv4.7

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9.0, y \cdot x, b\right) \cdot \frac{1}{z}} - \left(t \cdot a\right) \cdot 4.0}{c}\]

    if 1.4789431977947666e+196 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

    1. Initial program 43.1

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified24.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(9.0, y \cdot x, b\right)}{z} - \left(t \cdot a\right) \cdot 4.0}{c}}\]

    3. Taylor expanded around 0 22.5

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{a \cdot t}{c}}\]
    4. Using strategy rm

    5. Applied associate-/l*17.9

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{a \cdot t}{c}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.9

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{a \cdot t}{\color{blue}{1 \cdot c}}\]

    8. Applied times-frac12.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{t}{c}\right)}\]

    9. Simplified12.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \left(\color{blue}{a} \cdot \frac{t}{c}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -3.4143397402486294 \cdot 10^{-292}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - 4.0 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.4789431977947666 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \mathsf{fma}\left(9.0, y \cdot x, b\right) - \left(t \cdot a\right) \cdot 4.0}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \left(a \cdot \frac{t}{c}\right) \cdot 4.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))