Average Error: 20.7 → 6.8
Time: 24.4s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.77176064061640148 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \left(\left(a \cdot 4\right) \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{c}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.3080307062211267 \cdot 10^{258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{\frac{b}{z}}{c}\right) - \left(a \cdot 4\right) \cdot \frac{t}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \left(a \cdot 4\right) \cdot \frac{t}{c}\\ \end{array}\]
\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 = -\infty:\\
\;\;\;\;\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 1.77176064061640148 \cdot 10^{-115}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \left(\left(a \cdot 4\right) \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{c}}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.3080307062211267 \cdot 10^{258}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{\frac{b}{z}}{c}\right) - \left(a \cdot 4\right) \cdot \frac{t}{c}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r517137 = x;
        double r517138 = 9.0;
        double r517139 = r517137 * r517138;
        double r517140 = y;
        double r517141 = r517139 * r517140;
        double r517142 = z;
        double r517143 = 4.0;
        double r517144 = r517142 * r517143;
        double r517145 = t;
        double r517146 = r517144 * r517145;
        double r517147 = a;
        double r517148 = r517146 * r517147;
        double r517149 = r517141 - r517148;
        double r517150 = b;
        double r517151 = r517149 + r517150;
        double r517152 = c;
        double r517153 = r517142 * r517152;
        double r517154 = r517151 / r517153;
        return r517154;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r517155 = x;
        double r517156 = 9.0;
        double r517157 = r517155 * r517156;
        double r517158 = y;
        double r517159 = r517157 * r517158;
        double r517160 = -inf.0;
        bool r517161 = r517159 <= r517160;
        double r517162 = z;
        double r517163 = r517155 / r517162;
        double r517164 = c;
        double r517165 = r517158 / r517164;
        double r517166 = r517163 * r517165;
        double r517167 = b;
        double r517168 = r517162 * r517164;
        double r517169 = r517167 / r517168;
        double r517170 = fma(r517166, r517156, r517169);
        double r517171 = a;
        double r517172 = 4.0;
        double r517173 = r517171 * r517172;
        double r517174 = t;
        double r517175 = r517173 * r517174;
        double r517176 = r517175 / r517164;
        double r517177 = r517170 - r517176;
        double r517178 = 1.7717606406164015e-115;
        bool r517179 = r517159 <= r517178;
        double r517180 = r517155 * r517158;
        double r517181 = r517180 / r517168;
        double r517182 = fma(r517181, r517156, r517169);
        double r517183 = cbrt(r517174);
        double r517184 = r517183 * r517183;
        double r517185 = cbrt(r517164);
        double r517186 = r517185 * r517185;
        double r517187 = r517184 / r517186;
        double r517188 = r517173 * r517187;
        double r517189 = r517183 / r517185;
        double r517190 = r517188 * r517189;
        double r517191 = r517182 - r517190;
        double r517192 = 1.3080307062211267e+258;
        bool r517193 = r517159 <= r517192;
        double r517194 = r517167 / r517162;
        double r517195 = r517194 / r517164;
        double r517196 = fma(r517181, r517156, r517195);
        double r517197 = r517174 / r517164;
        double r517198 = r517173 * r517197;
        double r517199 = r517196 - r517198;
        double r517200 = r517170 - r517198;
        double r517201 = r517193 ? r517199 : r517200;
        double r517202 = r517179 ? r517191 : r517201;
        double r517203 = r517161 ? r517177 : r517202;
        return r517203;
}

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

Original20.7
Target14.9
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.1001567408041049 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* (* x 9.0) y) < -inf.0

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around 0 64.0

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

    4. Simplified64.0

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

    6. Applied times-frac11.1

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

    if -inf.0 < (* (* x 9.0) y) < 1.7717606406164015e-115

    1. Initial program 17.3

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

    2. Simplified9.1

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

    3. Taylor expanded around 0 8.1

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

    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 *-un-lft-identity8.1

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

    7. Applied times-frac7.4

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

    8. Simplified7.4

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

    10. Applied add-cube-cbrt7.8

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

    11. Applied add-cube-cbrt7.9

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

    12. Applied times-frac7.9

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

    13. Applied associate-*r*5.1

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

    if 1.7717606406164015e-115 < (* (* x 9.0) y) < 1.3080307062211267e+258

    1. Initial program 17.9

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

    2. Simplified10.1

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

    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 *-un-lft-identity8.1

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

    7. Applied times-frac7.3

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

    8. Simplified7.3

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

    10. Applied associate-/r*9.6

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

    if 1.3080307062211267e+258 < (* (* x 9.0) y)

    1. Initial program 50.9

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

    2. Simplified48.8

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

    3. Taylor expanded around 0 46.8

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

    4. Simplified46.8

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

    6. Applied *-un-lft-identity46.8

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

    7. Applied times-frac46.0

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

    8. Simplified46.0

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

    10. Applied times-frac10.3

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.77176064061640148 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \left(\left(a \cdot 4\right) \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{c}}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.3080307062211267 \cdot 10^{258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{\frac{b}{z}}{c}\right) - \left(a \cdot 4\right) \cdot \frac{t}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \left(a \cdot 4\right) \cdot \frac{t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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)))