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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.988825475455037498549084748882071237206 \cdot 10^{-159} \lor \neg \left(x \le 8.590299919376646895424917696537023407421 \cdot 10^{46}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

\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 -3.988825475455037498549084748882071237206 \cdot 10^{-159} \lor \neg \left(x \le 8.590299919376646895424917696537023407421 \cdot 10^{46}\right):\\
\;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\

\end{array}

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r647399 = x;
        double r647400 = y;
        double r647401 = z;
        double r647402 = r647400 * r647401;
        double r647403 = t;
        double r647404 = a;
        double r647405 = r647403 * r647404;
        double r647406 = r647402 - r647405;
        double r647407 = r647399 * r647406;
        double r647408 = b;
        double r647409 = c;
        double r647410 = r647409 * r647401;
        double r647411 = i;
        double r647412 = r647411 * r647404;
        double r647413 = r647410 - r647412;
        double r647414 = r647408 * r647413;
        double r647415 = r647407 - r647414;
        double r647416 = j;
        double r647417 = r647409 * r647403;
        double r647418 = r647411 * r647400;
        double r647419 = r647417 - r647418;
        double r647420 = r647416 * r647419;
        double r647421 = r647415 + r647420;
        return r647421;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r647422 = x;
        double r647423 = -3.9888254754550375e-159;
        bool r647424 = r647422 <= r647423;
        double r647425 = 8.590299919376647e+46;
        bool r647426 = r647422 <= r647425;
        double r647427 = !r647426;
        bool r647428 = r647424 || r647427;
        double r647429 = cbrt(r647422);
        double r647430 = r647429 * r647429;
        double r647431 = y;
        double r647432 = z;
        double r647433 = r647431 * r647432;
        double r647434 = r647429 * r647433;
        double r647435 = r647430 * r647434;
        double r647436 = t;
        double r647437 = a;
        double r647438 = r647436 * r647437;
        double r647439 = -r647438;
        double r647440 = r647422 * r647439;
        double r647441 = r647435 + r647440;
        double r647442 = b;
        double r647443 = c;
        double r647444 = r647443 * r647432;
        double r647445 = i;
        double r647446 = r647445 * r647437;
        double r647447 = r647444 - r647446;
        double r647448 = r647442 * r647447;
        double r647449 = r647441 - r647448;
        double r647450 = j;
        double r647451 = r647443 * r647436;
        double r647452 = r647445 * r647431;
        double r647453 = r647451 - r647452;
        double r647454 = r647450 * r647453;
        double r647455 = r647449 + r647454;
        double r647456 = r647422 * r647431;
        double r647457 = r647456 * r647432;
        double r647458 = -1.0;
        double r647459 = r647422 * r647436;
        double r647460 = r647437 * r647459;
        double r647461 = r647458 * r647460;
        double r647462 = r647457 + r647461;
        double r647463 = r647462 - r647448;
        double r647464 = cbrt(r647454);
        double r647465 = r647464 * r647464;
        double r647466 = r647465 * r647464;
        double r647467 = r647463 + r647466;
        double r647468 = r647428 ? r647455 : r647467;
        return r647468;
}

Target

Original	12.4
Target	16.1
Herbie	10.0

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

Derivation

Split input into 2 regimes
if x < -3.9888254754550375e-159 or 8.590299919376647e+46 < x
1. Initial program 9.2
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Using strategy rm
3. Applied sub-neg9.2
  \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z + \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied distribute-lft-in9.2
  \[\leadsto \left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
5. Using strategy rm
6. Applied add-cube-cbrt9.4
  \[\leadsto \left(\left(\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
7. Applied associate-*l*9.4
  \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
if -3.9888254754550375e-159 < x < 8.590299919376647e+46
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
  \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z + \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
4. Applied distribute-lft-in15.6
  \[\leadsto \left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
5. Using strategy rm
6. Applied associate-*r*13.0
  \[\leadsto \left(\left(\color{blue}{\left(x \cdot y\right) \cdot z} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
7. Taylor expanded around inf 10.3
  \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
8. Using strategy rm
9. Applied add-cube-cbrt10.6
  \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}}\]
Recombined 2 regimes into one program.
Final simplification10.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.988825475455037498549084748882071237206 \cdot 10^{-159} \lor \neg \left(x \le 8.590299919376646895424917696537023407421 \cdot 10^{46}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + -1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))))

Error

Try it out

Target

Derivation

`if x < -3.9888254754550375e-159 or 8.590299919376647e+46 < x`

`if -3.9888254754550375e-159 < x < 8.590299919376647e+46`

Reproduce

In
Out