Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.7 → 10.4

Time: 19.2s

Precision: 64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.071803987356108646236154501276931761132 \cdot 10^{127} \lor \neg \left(t \le 8.095220835801230557633946104079438724776 \cdot 10^{223}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r506475 = x;
        double r506476 = y;
        double r506477 = r506475 + r506476;
        double r506478 = z;
        double r506479 = t;
        double r506480 = r506478 - r506479;
        double r506481 = r506480 * r506476;
        double r506482 = a;
        double r506483 = r506482 - r506479;
        double r506484 = r506481 / r506483;
        double r506485 = r506477 - r506484;
        return r506485;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r506486 = t;
        double r506487 = -6.071803987356109e+127;
        bool r506488 = r506486 <= r506487;
        double r506489 = 8.095220835801231e+223;
        bool r506490 = r506486 <= r506489;
        double r506491 = !r506490;
        bool r506492 = r506488 || r506491;
        double r506493 = z;
        double r506494 = y;
        double r506495 = r506493 * r506494;
        double r506496 = r506495 / r506486;
        double r506497 = x;
        double r506498 = r506496 + r506497;
        double r506499 = r506497 + r506494;
        double r506500 = r506493 - r506486;
        double r506501 = a;
        double r506502 = r506501 - r506486;
        double r506503 = cbrt(r506502);
        double r506504 = r506503 * r506503;
        double r506505 = r506500 / r506504;
        double r506506 = r506494 / r506503;
        double r506507 = r506505 * r506506;
        double r506508 = cbrt(r506507);
        double r506509 = r506508 * r506508;
        double r506510 = cbrt(r506505);
        double r506511 = cbrt(r506506);
        double r506512 = r506511 * r506511;
        double r506513 = cbrt(r506512);
        double r506514 = cbrt(r506511);
        double r506515 = r506513 * r506514;
        double r506516 = r506510 * r506515;
        double r506517 = r506509 * r506516;
        double r506518 = r506499 - r506517;
        double r506519 = r506492 ? r506498 : r506518;
        return r506519;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.7
Target	8.4
Herbie	10.4

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -6.071803987356109e+127 or 8.095220835801231e+223 < t

   1. Initial program 33.7

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

   2. Taylor expanded around inf 17.7

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

   if -6.071803987356109e+127 < t < 8.095220835801231e+223

   1. Initial program 11.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 11.6

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

   4. Applied times-frac 8.0

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 8.1

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}}\]
   7. Using strategy rm

   8. Applied cbrt-prod 8.1

      \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 8.1

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

   11. Applied cbrt-prod 8.1

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.071803987356108646236154501276931761132 \cdot 10^{127} \lor \neg \left(t \le 8.095220835801230557633946104079438724776 \cdot 10^{223}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.47542934445772333e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))