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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.532769285011378 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + c \cdot \left(z \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) - \left(b \cdot \left(c \cdot z\right) + \left(\left(-b\right) \cdot a\right) \cdot i\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

\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}\;i \le 6.532769285011378 \cdot 10^{-44}:\\
\;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + c \cdot \left(z \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26224088 = x;
        double r26224089 = y;
        double r26224090 = z;
        double r26224091 = r26224089 * r26224090;
        double r26224092 = t;
        double r26224093 = a;
        double r26224094 = r26224092 * r26224093;
        double r26224095 = r26224091 - r26224094;
        double r26224096 = r26224088 * r26224095;
        double r26224097 = b;
        double r26224098 = c;
        double r26224099 = r26224098 * r26224090;
        double r26224100 = i;
        double r26224101 = r26224100 * r26224093;
        double r26224102 = r26224099 - r26224101;
        double r26224103 = r26224097 * r26224102;
        double r26224104 = r26224096 - r26224103;
        double r26224105 = j;
        double r26224106 = r26224098 * r26224092;
        double r26224107 = r26224100 * r26224089;
        double r26224108 = r26224106 - r26224107;
        double r26224109 = r26224105 * r26224108;
        double r26224110 = r26224104 + r26224109;
        return r26224110;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26224111 = i;
        double r26224112 = 6.532769285011378e-44;
        bool r26224113 = r26224111 <= r26224112;
        double r26224114 = y;
        double r26224115 = z;
        double r26224116 = r26224114 * r26224115;
        double r26224117 = t;
        double r26224118 = a;
        double r26224119 = r26224117 * r26224118;
        double r26224120 = r26224116 - r26224119;
        double r26224121 = x;
        double r26224122 = r26224120 * r26224121;
        double r26224123 = b;
        double r26224124 = -r26224123;
        double r26224125 = r26224111 * r26224118;
        double r26224126 = r26224124 * r26224125;
        double r26224127 = c;
        double r26224128 = r26224115 * r26224123;
        double r26224129 = r26224127 * r26224128;
        double r26224130 = r26224126 + r26224129;
        double r26224131 = r26224122 - r26224130;
        double r26224132 = j;
        double r26224133 = r26224127 * r26224117;
        double r26224134 = r26224114 * r26224111;
        double r26224135 = r26224133 - r26224134;
        double r26224136 = r26224132 * r26224135;
        double r26224137 = r26224131 + r26224136;
        double r26224138 = cbrt(r26224122);
        double r26224139 = r26224138 * r26224138;
        double r26224140 = r26224138 * r26224139;
        double r26224141 = r26224127 * r26224115;
        double r26224142 = r26224123 * r26224141;
        double r26224143 = r26224124 * r26224118;
        double r26224144 = r26224143 * r26224111;
        double r26224145 = r26224142 + r26224144;
        double r26224146 = r26224140 - r26224145;
        double r26224147 = r26224146 + r26224136;
        double r26224148 = r26224113 ? r26224137 : r26224147;
        return r26224148;
}

Target

Original	11.9
Target	15.6
Herbie	11.5

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

Derivation

Split input into 2 regimes
if i < 6.532769285011378e-44
1. Initial program 10.8
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg10.8
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z + \left(-i \cdot a\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied distribute-rgt-in10.8
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(\left(c \cdot z\right) \cdot b + \left(-i \cdot a\right) \cdot b\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
5. Using strategy rm
6. Applied associate-*l*11.2
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{c \cdot \left(z \cdot b\right)} + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
if 6.532769285011378e-44 < i
1. Initial program 15.6
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z + \left(-i \cdot a\right)\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied distribute-rgt-in15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(\left(c \cdot z\right) \cdot b + \left(-i \cdot a\right) \cdot b\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
5. Using strategy rm
6. Applied distribute-lft-neg-in15.6
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(c \cdot z\right) \cdot b + \color{blue}{\left(\left(-i\right) \cdot a\right)} \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
7. Applied associate-*l*12.5
  \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(c \cdot z\right) \cdot b + \color{blue}{\left(-i\right) \cdot \left(a \cdot b\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
8. Using strategy rm
9. Applied add-cube-cbrt12.7
  \[\leadsto \left(\color{blue}{\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}} - \left(\left(c \cdot z\right) \cdot b + \left(-i\right) \cdot \left(a \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Recombined 2 regimes into one program.
Final simplification11.5
\[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.532769285011378 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + c \cdot \left(z \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) - \left(b \cdot \left(c \cdot z\right) + \left(\left(-b\right) \cdot a\right) \cdot i\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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)))))

Error

Try it out

Target

Derivation

`if i < 6.532769285011378e-44`

`if 6.532769285011378e-44 < i`

Reproduce

In
Out