Average Error: 20.3 → 3.9
Time: 27.5s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.036832002696555609781851781472359532756 \cdot 10^{290}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -5.61990937755884756231922496601490270958 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 5.44177297429320749590194194951108884969 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}}\right) - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 4.40589149168671384163062716676752837828 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]
\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}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.036832002696555609781851781472359532756 \cdot 10^{290}:\\
\;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9\right) - 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}{c \cdot z} \le -5.61990937755884756231922496601490270958 \cdot 10^{-292}:\\
\;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 5.44177297429320749590194194951108884969 \cdot 10^{-127}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}}\right) - \left(a \cdot 4\right) \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}{c \cdot z} \le 4.40589149168671384163062716676752837828 \cdot 10^{307}:\\
\;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r30678908 = x;
        double r30678909 = 9.0;
        double r30678910 = r30678908 * r30678909;
        double r30678911 = y;
        double r30678912 = r30678910 * r30678911;
        double r30678913 = z;
        double r30678914 = 4.0;
        double r30678915 = r30678913 * r30678914;
        double r30678916 = t;
        double r30678917 = r30678915 * r30678916;
        double r30678918 = a;
        double r30678919 = r30678917 * r30678918;
        double r30678920 = r30678912 - r30678919;
        double r30678921 = b;
        double r30678922 = r30678920 + r30678921;
        double r30678923 = c;
        double r30678924 = r30678913 * r30678923;
        double r30678925 = r30678922 / r30678924;
        return r30678925;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r30678926 = x;
        double r30678927 = 9.0;
        double r30678928 = r30678926 * r30678927;
        double r30678929 = y;
        double r30678930 = r30678928 * r30678929;
        double r30678931 = z;
        double r30678932 = 4.0;
        double r30678933 = r30678931 * r30678932;
        double r30678934 = t;
        double r30678935 = r30678933 * r30678934;
        double r30678936 = a;
        double r30678937 = r30678935 * r30678936;
        double r30678938 = r30678930 - r30678937;
        double r30678939 = b;
        double r30678940 = r30678938 + r30678939;
        double r30678941 = c;
        double r30678942 = r30678941 * r30678931;
        double r30678943 = r30678940 / r30678942;
        double r30678944 = -1.0368320026965556e+290;
        bool r30678945 = r30678943 <= r30678944;
        double r30678946 = r30678939 / r30678942;
        double r30678947 = r30678942 / r30678929;
        double r30678948 = r30678926 / r30678947;
        double r30678949 = r30678948 * r30678927;
        double r30678950 = r30678946 + r30678949;
        double r30678951 = r30678934 * r30678936;
        double r30678952 = r30678951 / r30678941;
        double r30678953 = r30678932 * r30678952;
        double r30678954 = r30678950 - r30678953;
        double r30678955 = -5.619909377558848e-292;
        bool r30678956 = r30678943 <= r30678955;
        double r30678957 = 5.4417729742932075e-127;
        bool r30678958 = r30678943 <= r30678957;
        double r30678959 = fma(r30678928, r30678929, r30678939);
        double r30678960 = r30678959 / r30678931;
        double r30678961 = cbrt(r30678960);
        double r30678962 = r30678961 * r30678961;
        double r30678963 = r30678961 * r30678962;
        double r30678964 = r30678936 * r30678932;
        double r30678965 = r30678964 * r30678934;
        double r30678966 = r30678963 - r30678965;
        double r30678967 = r30678966 / r30678941;
        double r30678968 = 4.405891491686714e+307;
        bool r30678969 = r30678943 <= r30678968;
        double r30678970 = r30678929 / r30678941;
        double r30678971 = r30678926 / r30678931;
        double r30678972 = r30678970 * r30678971;
        double r30678973 = r30678927 * r30678972;
        double r30678974 = r30678973 + r30678946;
        double r30678975 = r30678936 / r30678941;
        double r30678976 = r30678975 * r30678934;
        double r30678977 = r30678932 * r30678976;
        double r30678978 = r30678974 - r30678977;
        double r30678979 = r30678969 ? r30678943 : r30678978;
        double r30678980 = r30678958 ? r30678967 : r30678979;
        double r30678981 = r30678956 ? r30678943 : r30678980;
        double r30678982 = r30678945 ? r30678954 : r30678981;
        return r30678982;
}

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

Original20.3
Target14.1
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.0368320026965556e+290

    1. Initial program 55.5

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

    2. Simplified23.1

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

    3. Taylor expanded around 0 27.0

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

    5. Applied associate-/l*16.9

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

    if -1.0368320026965556e+290 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -5.619909377558848e-292 or 5.4417729742932075e-127 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 4.405891491686714e+307

    1. Initial program 0.6

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

    if -5.619909377558848e-292 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 5.4417729742932075e-127

    1. Initial program 27.4

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

    2. Simplified0.6

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

    4. Applied add-cube-cbrt1.1

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

    if 4.405891491686714e+307 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

    1. Initial program 63.8

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

    2. Simplified28.5

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

    3. Taylor expanded around 0 32.0

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

    5. Applied *-un-lft-identity32.0

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

    6. Applied times-frac27.5

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

    7. Simplified27.5

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

    9. Applied times-frac11.0

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.036832002696555609781851781472359532756 \cdot 10^{290}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -5.61990937755884756231922496601490270958 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 5.44177297429320749590194194951108884969 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \left(\sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z}}\right) - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 4.40589149168671384163062716676752837828 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))