Average Error: 11.3 → 11.2
Time: 32.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.645805544531443 \cdot 10^{-232}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\ \mathbf{elif}\;x \le 2.7793681624166355 \cdot 10^{-219}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot a\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.645805544531443 \cdot 10^{-232}:\\
\;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\

\mathbf{elif}\;x \le 2.7793681624166355 \cdot 10^{-219}:\\
\;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot a\right) \cdot \left(-b\right)\\

\mathbf{else}:\\
\;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r28305083 = x;
        double r28305084 = y;
        double r28305085 = z;
        double r28305086 = r28305084 * r28305085;
        double r28305087 = t;
        double r28305088 = a;
        double r28305089 = r28305087 * r28305088;
        double r28305090 = r28305086 - r28305089;
        double r28305091 = r28305083 * r28305090;
        double r28305092 = b;
        double r28305093 = c;
        double r28305094 = r28305093 * r28305085;
        double r28305095 = i;
        double r28305096 = r28305095 * r28305088;
        double r28305097 = r28305094 - r28305096;
        double r28305098 = r28305092 * r28305097;
        double r28305099 = r28305091 - r28305098;
        double r28305100 = j;
        double r28305101 = r28305093 * r28305087;
        double r28305102 = r28305095 * r28305084;
        double r28305103 = r28305101 - r28305102;
        double r28305104 = r28305100 * r28305103;
        double r28305105 = r28305099 + r28305104;
        return r28305105;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r28305106 = x;
        double r28305107 = -3.645805544531443e-232;
        bool r28305108 = r28305106 <= r28305107;
        double r28305109 = c;
        double r28305110 = t;
        double r28305111 = r28305109 * r28305110;
        double r28305112 = i;
        double r28305113 = y;
        double r28305114 = r28305112 * r28305113;
        double r28305115 = r28305111 - r28305114;
        double r28305116 = j;
        double r28305117 = r28305115 * r28305116;
        double r28305118 = z;
        double r28305119 = r28305113 * r28305118;
        double r28305120 = a;
        double r28305121 = r28305120 * r28305110;
        double r28305122 = r28305119 - r28305121;
        double r28305123 = r28305122 * r28305106;
        double r28305124 = b;
        double r28305125 = r28305118 * r28305109;
        double r28305126 = r28305112 * r28305120;
        double r28305127 = r28305125 - r28305126;
        double r28305128 = r28305124 * r28305127;
        double r28305129 = cbrt(r28305128);
        double r28305130 = cbrt(r28305127);
        double r28305131 = cbrt(r28305124);
        double r28305132 = r28305130 * r28305131;
        double r28305133 = r28305129 * r28305132;
        double r28305134 = r28305129 * r28305133;
        double r28305135 = r28305123 - r28305134;
        double r28305136 = r28305117 + r28305135;
        double r28305137 = 2.7793681624166355e-219;
        bool r28305138 = r28305106 <= r28305137;
        double r28305139 = -r28305124;
        double r28305140 = r28305127 * r28305139;
        double r28305141 = r28305117 + r28305140;
        double r28305142 = r28305138 ? r28305141 : r28305136;
        double r28305143 = r28305108 ? r28305136 : r28305142;
        return r28305143;
}

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

Bits error versus i

Bits error versus j

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.3
Target14.9
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;t \lt -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.645805544531443e-232 or 2.7793681624166355e-219 < x

    1. Initial program 10.1

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt10.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    4. Using strategy rm

    5. Applied cbrt-prod10.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    if -3.645805544531443e-232 < x < 2.7793681624166355e-219

    1. Initial program 16.5

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    2. Taylor expanded around 0 15.4

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.645805544531443 \cdot 10^{-232}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\ \mathbf{elif}\;x \le 2.7793681624166355 \cdot 10^{-219}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(z \cdot c - i \cdot a\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{z \cdot c - i \cdot a} \cdot \sqrt[3]{b}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))