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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]

\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}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\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 r601532 = x;
        double r601533 = y;
        double r601534 = z;
        double r601535 = r601533 * r601534;
        double r601536 = t;
        double r601537 = a;
        double r601538 = r601536 * r601537;
        double r601539 = r601535 - r601538;
        double r601540 = r601532 * r601539;
        double r601541 = b;
        double r601542 = c;
        double r601543 = r601542 * r601534;
        double r601544 = i;
        double r601545 = r601544 * r601537;
        double r601546 = r601543 - r601545;
        double r601547 = r601541 * r601546;
        double r601548 = r601540 - r601547;
        double r601549 = j;
        double r601550 = r601542 * r601536;
        double r601551 = r601544 * r601533;
        double r601552 = r601550 - r601551;
        double r601553 = r601549 * r601552;
        double r601554 = r601548 + r601553;
        return r601554;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r601555 = y;
        double r601556 = -2.690019024448134e+149;
        bool r601557 = r601555 <= r601556;
        double r601558 = 6.851977512719908e+134;
        bool r601559 = r601555 <= r601558;
        double r601560 = !r601559;
        bool r601561 = r601557 || r601560;
        double r601562 = i;
        double r601563 = a;
        double r601564 = r601562 * r601563;
        double r601565 = c;
        double r601566 = z;
        double r601567 = r601565 * r601566;
        double r601568 = r601564 - r601567;
        double r601569 = b;
        double r601570 = x;
        double r601571 = r601570 * r601566;
        double r601572 = j;
        double r601573 = r601562 * r601572;
        double r601574 = r601571 - r601573;
        double r601575 = r601555 * r601574;
        double r601576 = t;
        double r601577 = r601570 * r601576;
        double r601578 = r601563 * r601577;
        double r601579 = r601575 - r601578;
        double r601580 = fma(r601568, r601569, r601579);
        double r601581 = r601565 * r601576;
        double r601582 = r601562 * r601555;
        double r601583 = r601581 - r601582;
        double r601584 = cbrt(r601570);
        double r601585 = r601584 * r601584;
        double r601586 = r601555 * r601566;
        double r601587 = r601576 * r601563;
        double r601588 = r601586 - r601587;
        double r601589 = r601584 * r601588;
        double r601590 = r601585 * r601589;
        double r601591 = fma(r601572, r601583, r601590);
        double r601592 = fma(r601568, r601569, r601591);
        double r601593 = r601561 ? r601580 : r601592;
        return r601593;
}

Target

Original	12.3
Target	16.3
Herbie	10.4

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if y < -2.690019024448134e+149 or 6.851977512719908e+134 < y
1. Initial program 23.4
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Simplified23.4
  \[\leadsto \color{blue}{\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt23.6
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*23.6
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
6. Taylor expanded around inf 25.0
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \color{blue}{x \cdot \left(z \cdot y\right) - \left(i \cdot \left(j \cdot y\right) + a \cdot \left(x \cdot t\right)\right)}\right)\]
7. Simplified10.5
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \color{blue}{y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)}\right)\]
if -2.690019024448134e+149 < y < 6.851977512719908e+134
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. Simplified10.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt10.4
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*10.3
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
Recombined 2 regimes into one program.
Final simplification10.4
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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 -8.120978919195912e-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)))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if y < -2.690019024448134e+149 or 6.851977512719908e+134 < y`

`if -2.690019024448134e+149 < y < 6.851977512719908e+134`

Reproduce