Average Error: 6.7 → 2.4
Time: 16.7s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[-\frac{2}{\left(\left(y - t\right) \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)\right) \cdot \left(-\frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
-\frac{2}{\left(\left(y - t\right) \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)\right) \cdot \left(-\frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)}
double f(double x, double y, double z, double t) {
        double r26167389 = x;
        double r26167390 = 2.0;
        double r26167391 = r26167389 * r26167390;
        double r26167392 = y;
        double r26167393 = z;
        double r26167394 = r26167392 * r26167393;
        double r26167395 = t;
        double r26167396 = r26167395 * r26167393;
        double r26167397 = r26167394 - r26167396;
        double r26167398 = r26167391 / r26167397;
        return r26167398;
}
↓
double f(double x, double y, double z, double t) {
        double r26167399 = 2.0;
        double r26167400 = y;
        double r26167401 = t;
        double r26167402 = r26167400 - r26167401;
        double r26167403 = z;
        double r26167404 = cbrt(r26167403);
        double r26167405 = x;
        double r26167406 = cbrt(r26167405);
        double r26167407 = r26167404 / r26167406;
        double r26167408 = r26167407 * r26167407;
        double r26167409 = r26167402 * r26167408;
        double r26167410 = -r26167407;
        double r26167411 = r26167409 * r26167410;
        double r26167412 = r26167399 / r26167411;
        double r26167413 = -r26167412;
        return r26167413;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.7
Target2.2
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 6.7

    \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

  2. Simplified5.9

    \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot \left(y - t\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt6.6

    \[\leadsto \frac{2}{\frac{z}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \cdot \left(y - t\right)}\]

  5. Applied add-cube-cbrt6.8

    \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \left(y - t\right)}\]

  6. Applied times-frac6.8

    \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)} \cdot \left(y - t\right)}\]

  7. Applied associate-*l*2.4

    \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \left(y - t\right)\right)}}\]
  8. Using strategy rm

  9. Applied frac-2neg2.4

    \[\leadsto \color{blue}{\frac{-2}{-\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \left(y - t\right)\right)}}\]

  10. Simplified2.4

    \[\leadsto \frac{-2}{\color{blue}{-\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \left(\left(y - t\right) \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)\right)}}\]

  11. Final simplification2.4

    \[\leadsto -\frac{2}{\left(\left(y - t\right) \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)\right) \cdot \left(-\frac{\sqrt[3]{z}}{\sqrt[3]{x}}\right)}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))