Average Error: 12.3 → 11.3
Time: 27.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.973496614217344094601403267363420899231 \cdot 10^{-55} \lor \neg \left(z \le 1.751377172133955651818986187295260027638 \cdot 10^{-61}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]
\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}\;z \le -2.973496614217344094601403267363420899231 \cdot 10^{-55} \lor \neg \left(z \le 1.751377172133955651818986187295260027638 \cdot 10^{-61}\right):\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-\left(i \cdot j\right) \cdot y\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 r383054 = x;
        double r383055 = y;
        double r383056 = z;
        double r383057 = r383055 * r383056;
        double r383058 = t;
        double r383059 = a;
        double r383060 = r383058 * r383059;
        double r383061 = r383057 - r383060;
        double r383062 = r383054 * r383061;
        double r383063 = b;
        double r383064 = c;
        double r383065 = r383064 * r383056;
        double r383066 = i;
        double r383067 = r383066 * r383059;
        double r383068 = r383065 - r383067;
        double r383069 = r383063 * r383068;
        double r383070 = r383062 - r383069;
        double r383071 = j;
        double r383072 = r383064 * r383058;
        double r383073 = r383066 * r383055;
        double r383074 = r383072 - r383073;
        double r383075 = r383071 * r383074;
        double r383076 = r383070 + r383075;
        return r383076;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r383077 = z;
        double r383078 = -2.973496614217344e-55;
        bool r383079 = r383077 <= r383078;
        double r383080 = 1.7513771721339557e-61;
        bool r383081 = r383077 <= r383080;
        double r383082 = !r383081;
        bool r383083 = r383079 || r383082;
        double r383084 = x;
        double r383085 = y;
        double r383086 = r383085 * r383077;
        double r383087 = t;
        double r383088 = a;
        double r383089 = r383087 * r383088;
        double r383090 = r383086 - r383089;
        double r383091 = r383084 * r383090;
        double r383092 = b;
        double r383093 = c;
        double r383094 = r383092 * r383093;
        double r383095 = r383077 * r383094;
        double r383096 = i;
        double r383097 = r383096 * r383088;
        double r383098 = -r383097;
        double r383099 = r383098 * r383092;
        double r383100 = r383095 + r383099;
        double r383101 = r383091 - r383100;
        double r383102 = j;
        double r383103 = r383102 * r383093;
        double r383104 = r383103 * r383087;
        double r383105 = r383102 * r383085;
        double r383106 = r383096 * r383105;
        double r383107 = -r383106;
        double r383108 = r383104 + r383107;
        double r383109 = r383101 + r383108;
        double r383110 = r383086 * r383084;
        double r383111 = r383084 * r383087;
        double r383112 = r383088 * r383111;
        double r383113 = -r383112;
        double r383114 = r383110 + r383113;
        double r383115 = r383093 * r383077;
        double r383116 = r383115 - r383097;
        double r383117 = r383092 * r383116;
        double r383118 = r383114 - r383117;
        double r383119 = r383096 * r383102;
        double r383120 = r383119 * r383085;
        double r383121 = -r383120;
        double r383122 = r383104 + r383121;
        double r383123 = r383118 + r383122;
        double r383124 = r383083 ? r383109 : r383123;
        return r383124;
}

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

Original12.3
Target16.1
Herbie11.3
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.973496614217344e-55 or 1.7513771721339557e-61 < z

    1. Initial program 15.3

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

    3. Applied sub-neg15.3

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

    4. Applied distribute-lft-in15.3

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

    5. Simplified15.8

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

    7. Applied associate-*r*15.9

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

    9. Applied sub-neg15.9

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

    10. Applied distribute-lft-in15.9

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

    11. Simplified12.7

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

    12. Simplified12.7

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

    if -2.973496614217344e-55 < z < 1.7513771721339557e-61

    1. Initial program 9.5

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

    3. Applied sub-neg9.5

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

    4. Applied distribute-lft-in9.5

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

    5. Simplified9.7

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

    7. Applied associate-*r*10.0

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

    9. Applied associate-*r*9.3

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

    11. Applied sub-neg9.3

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

    12. Applied distribute-lft-in9.3

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

    13. Simplified9.3

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

    14. Simplified10.0

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.973496614217344094601403267363420899231 \cdot 10^{-55} \lor \neg \left(z \le 1.751377172133955651818986187295260027638 \cdot 10^{-61}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.1209789191959122e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.7125538182184851e-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 -7.63353334603158369e-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)))))