Average Error: 7.1 → 1.9
Time: 1.0m
Precision: 64
\[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
↓
\[\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\frac{y - t}{2.0}}{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}}\]
\frac{x \cdot 2.0}{y \cdot z - t \cdot z}
↓
\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\frac{y - t}{2.0}}{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}}
double f(double x, double y, double z, double t) {
        double r24120363 = x;
        double r24120364 = 2.0;
        double r24120365 = r24120363 * r24120364;
        double r24120366 = y;
        double r24120367 = z;
        double r24120368 = r24120366 * r24120367;
        double r24120369 = t;
        double r24120370 = r24120369 * r24120367;
        double r24120371 = r24120368 - r24120370;
        double r24120372 = r24120365 / r24120371;
        return r24120372;
}
↓
double f(double x, double y, double z, double t) {
        double r24120373 = x;
        double r24120374 = cbrt(r24120373);
        double r24120375 = r24120374 * r24120374;
        double r24120376 = z;
        double r24120377 = cbrt(r24120376);
        double r24120378 = r24120377 * r24120377;
        double r24120379 = r24120375 / r24120378;
        double r24120380 = y;
        double r24120381 = t;
        double r24120382 = r24120380 - r24120381;
        double r24120383 = 2.0;
        double r24120384 = r24120382 / r24120383;
        double r24120385 = r24120374 / r24120377;
        double r24120386 = r24120384 / r24120385;
        double r24120387 = r24120379 / r24120386;
        return r24120387;
}

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

Original7.1
Target2.1
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \mathbf{elif}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2.0}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \end{array}\]

Derivation

  1. Initial program 7.1

    \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]

  2. Simplified5.6

    \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{y - t}{2.0}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt6.3

    \[\leadsto \frac{\frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}{\frac{y - t}{2.0}}\]

  5. Applied add-cube-cbrt6.4

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\frac{y - t}{2.0}}\]

  6. Applied times-frac6.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}}}{\frac{y - t}{2.0}}\]

  7. Applied associate-/l*1.9

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\frac{y - t}{2.0}}{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}}}\]

  8. Final simplification1.9

    \[\leadsto \frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\frac{\frac{y - t}{2.0}}{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}}\]

Reproduce

herbie shell --seed 2019163 +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))))