Average Error: 8.1 → 4.9
Time: 17.5s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot 2 \le -1.009474070510834454191666288237777213198 \cdot 10^{203}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{a} \cdot y\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;a \cdot 2 \le -112410025309132171858390548480:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;a \cdot 2 \le 5.942825833029072965236157261173754109009 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -\left(9 \cdot t\right) \cdot z\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;a \cdot 2 \le -1.009474070510834454191666288237777213198 \cdot 10^{203}:\\
\;\;\;\;0.5 \cdot \left(\frac{x}{a} \cdot y\right) - 4.5 \cdot \frac{t \cdot z}{a}\\

\mathbf{elif}\;a \cdot 2 \le -112410025309132171858390548480:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\

\mathbf{elif}\;a \cdot 2 \le 5.942825833029072965236157261173754109009 \cdot 10^{-93}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y, -\left(9 \cdot t\right) \cdot z\right)}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r526112 = x;
        double r526113 = y;
        double r526114 = r526112 * r526113;
        double r526115 = z;
        double r526116 = 9.0;
        double r526117 = r526115 * r526116;
        double r526118 = t;
        double r526119 = r526117 * r526118;
        double r526120 = r526114 - r526119;
        double r526121 = a;
        double r526122 = 2.0;
        double r526123 = r526121 * r526122;
        double r526124 = r526120 / r526123;
        return r526124;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r526125 = a;
        double r526126 = 2.0;
        double r526127 = r526125 * r526126;
        double r526128 = -1.0094740705108345e+203;
        bool r526129 = r526127 <= r526128;
        double r526130 = 0.5;
        double r526131 = x;
        double r526132 = r526131 / r526125;
        double r526133 = y;
        double r526134 = r526132 * r526133;
        double r526135 = r526130 * r526134;
        double r526136 = 4.5;
        double r526137 = t;
        double r526138 = z;
        double r526139 = r526137 * r526138;
        double r526140 = r526139 / r526125;
        double r526141 = r526136 * r526140;
        double r526142 = r526135 - r526141;
        double r526143 = -1.1241002530913217e+29;
        bool r526144 = r526127 <= r526143;
        double r526145 = r526125 / r526133;
        double r526146 = r526131 / r526145;
        double r526147 = r526130 * r526146;
        double r526148 = cbrt(r526125);
        double r526149 = r526148 * r526148;
        double r526150 = r526137 / r526149;
        double r526151 = r526136 * r526150;
        double r526152 = r526138 / r526148;
        double r526153 = r526151 * r526152;
        double r526154 = r526147 - r526153;
        double r526155 = 5.942825833029073e-93;
        bool r526156 = r526127 <= r526155;
        double r526157 = 9.0;
        double r526158 = r526157 * r526137;
        double r526159 = r526158 * r526138;
        double r526160 = -r526159;
        double r526161 = fma(r526131, r526133, r526160);
        double r526162 = r526161 / r526127;
        double r526163 = r526136 * r526137;
        double r526164 = r526138 / r526125;
        double r526165 = r526163 * r526164;
        double r526166 = r526147 - r526165;
        double r526167 = r526156 ? r526162 : r526166;
        double r526168 = r526144 ? r526154 : r526167;
        double r526169 = r526129 ? r526142 : r526168;
        return r526169;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original8.1
Target5.8
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a 2.0) < -1.0094740705108345e+203

    1. Initial program 14.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 14.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied associate-/l*11.8

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
    5. Using strategy rm

    6. Applied associate-/r/11.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{a} \cdot y\right)} - 4.5 \cdot \frac{t \cdot z}{a}\]

    if -1.0094740705108345e+203 < (* a 2.0) < -1.1241002530913217e+29

    1. Initial program 11.6

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 11.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied associate-/l*8.6

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt8.9

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    7. Applied times-frac4.0

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]

    8. Applied associate-*r*4.0

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}}\]

    if -1.1241002530913217e+29 < (* a 2.0) < 5.942825833029073e-93

    1. Initial program 1.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
    2. Using strategy rm

    3. Applied fma-neg1.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]
    4. Using strategy rm

    5. Applied pow11.8

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot \color{blue}{{t}^{1}}\right)}{a \cdot 2}\]

    6. Applied pow11.8

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -\left(z \cdot \color{blue}{{9}^{1}}\right) \cdot {t}^{1}\right)}{a \cdot 2}\]

    7. Applied pow11.8

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -\left(\color{blue}{{z}^{1}} \cdot {9}^{1}\right) \cdot {t}^{1}\right)}{a \cdot 2}\]

    8. Applied pow-prod-down1.8

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{{\left(z \cdot 9\right)}^{1}} \cdot {t}^{1}\right)}{a \cdot 2}\]

    9. Applied pow-prod-down1.8

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -\color{blue}{{\left(\left(z \cdot 9\right) \cdot t\right)}^{1}}\right)}{a \cdot 2}\]

    10. Simplified1.7

      \[\leadsto \frac{\mathsf{fma}\left(x, y, -{\color{blue}{\left(\left(9 \cdot t\right) \cdot z\right)}}^{1}\right)}{a \cdot 2}\]

    if 5.942825833029073e-93 < (* a 2.0)

    1. Initial program 9.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 9.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied associate-/l*8.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity8.1

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

    7. Applied times-frac5.8

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

    8. Applied associate-*r*5.9

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

    9. Simplified5.9

      \[\leadsto 0.5 \cdot \frac{x}{\frac{a}{y}} - \color{blue}{\left(4.5 \cdot t\right)} \cdot \frac{z}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \le -1.009474070510834454191666288237777213198 \cdot 10^{203}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{a} \cdot y\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;a \cdot 2 \le -112410025309132171858390548480:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{elif}\;a \cdot 2 \le 5.942825833029072965236157261173754109009 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, -\left(9 \cdot t\right) \cdot z\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))