Average Error: 20.7 → 8.5
Time: 23.9s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.102451017909847122514360410827302725451 \cdot 10^{-121}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.49942252444617336175227318057610535812 \cdot 10^{-35}:\\ \;\;\;\;\left(\frac{9 \cdot \left(x \cdot y\right) + b}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 9.015864611909133506028130612168974390184 \cdot 10^{300}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]
\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}\;\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:\\
\;\;\;\;-4 \cdot \frac{t \cdot a}{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} \le -2.102451017909847122514360410827302725451 \cdot 10^{-121}:\\
\;\;\;\;\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{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 2.49942252444617336175227318057610535812 \cdot 10^{-35}:\\
\;\;\;\;\left(\frac{9 \cdot \left(x \cdot y\right) + b}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{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} \le 9.015864611909133506028130612168974390184 \cdot 10^{300}:\\
\;\;\;\;\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}:\\
\;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r544950 = x;
        double r544951 = 9.0;
        double r544952 = r544950 * r544951;
        double r544953 = y;
        double r544954 = r544952 * r544953;
        double r544955 = z;
        double r544956 = 4.0;
        double r544957 = r544955 * r544956;
        double r544958 = t;
        double r544959 = r544957 * r544958;
        double r544960 = a;
        double r544961 = r544959 * r544960;
        double r544962 = r544954 - r544961;
        double r544963 = b;
        double r544964 = r544962 + r544963;
        double r544965 = c;
        double r544966 = r544955 * r544965;
        double r544967 = r544964 / r544966;
        return r544967;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r544968 = x;
        double r544969 = 9.0;
        double r544970 = r544968 * r544969;
        double r544971 = y;
        double r544972 = r544970 * r544971;
        double r544973 = z;
        double r544974 = 4.0;
        double r544975 = r544973 * r544974;
        double r544976 = t;
        double r544977 = r544975 * r544976;
        double r544978 = a;
        double r544979 = r544977 * r544978;
        double r544980 = r544972 - r544979;
        double r544981 = b;
        double r544982 = r544980 + r544981;
        double r544983 = c;
        double r544984 = r544973 * r544983;
        double r544985 = r544982 / r544984;
        double r544986 = -inf.0;
        bool r544987 = r544985 <= r544986;
        double r544988 = -4.0;
        double r544989 = r544976 * r544978;
        double r544990 = r544989 / r544983;
        double r544991 = r544988 * r544990;
        double r544992 = -2.102451017909847e-121;
        bool r544993 = r544985 <= r544992;
        double r544994 = 2.4994225244461734e-35;
        bool r544995 = r544985 <= r544994;
        double r544996 = r544968 * r544971;
        double r544997 = r544969 * r544996;
        double r544998 = r544997 + r544981;
        double r544999 = r544998 / r544973;
        double r545000 = r544978 * r544974;
        double r545001 = r545000 * r544976;
        double r545002 = r544999 - r545001;
        double r545003 = 1.0;
        double r545004 = r545003 / r544983;
        double r545005 = r545002 * r545004;
        double r545006 = 9.015864611909134e+300;
        bool r545007 = r544985 <= r545006;
        double r545008 = r545007 ? r544985 : r544991;
        double r545009 = r544995 ? r545005 : r545008;
        double r545010 = r544993 ? r544985 : r545009;
        double r545011 = r544987 ? r544991 : r545010;
        return r545011;
}

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

Original20.7
Target14.6
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 9.015864611909134e+300 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

    1. Initial program 62.8

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

    2. Simplified25.6

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

    4. Applied clear-num25.6

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

    5. Taylor expanded around inf 31.3

      \[\leadsto \color{blue}{-4 \cdot \frac{t \cdot a}{c}}\]

    if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.102451017909847e-121 or 2.4994225244461734e-35 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 9.015864611909134e+300

    1. Initial program 0.7

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

    if -2.102451017909847e-121 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.4994225244461734e-35

    1. Initial program 20.6

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

    2. Simplified0.9

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

    4. Applied div-inv1.0

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

    6. Applied fma-udef1.0

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

    7. Simplified1.0

      \[\leadsto \left(\frac{\color{blue}{9 \cdot \left(x \cdot y\right)} + b}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.102451017909847122514360410827302725451 \cdot 10^{-121}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.49942252444617336175227318057610535812 \cdot 10^{-35}:\\ \;\;\;\;\left(\frac{9 \cdot \left(x \cdot y\right) + b}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 9.015864611909133506028130612168974390184 \cdot 10^{300}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)) -1.100156740804105e-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)))