Average Error: 6.8 → 2.7
Time: 6.4s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.9379949802427703 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \mathbf{elif}\;z \le 4.5360233662522261 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.9379949802427703 \cdot 10^{-128}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\

\mathbf{elif}\;z \le 4.5360233662522261 \cdot 10^{-37}:\\
\;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r612562 = x;
        double r612563 = 2.0;
        double r612564 = r612562 * r612563;
        double r612565 = y;
        double r612566 = z;
        double r612567 = r612565 * r612566;
        double r612568 = t;
        double r612569 = r612568 * r612566;
        double r612570 = r612567 - r612569;
        double r612571 = r612564 / r612570;
        return r612571;
}
↓
double f(double x, double y, double z, double t) {
        double r612572 = z;
        double r612573 = -6.93799498024277e-128;
        bool r612574 = r612572 <= r612573;
        double r612575 = x;
        double r612576 = 2.0;
        double r612577 = r612575 * r612576;
        double r612578 = y;
        double r612579 = t;
        double r612580 = r612578 - r612579;
        double r612581 = r612577 / r612580;
        double r612582 = r612581 / r612572;
        double r612583 = 4.536023366252226e-37;
        bool r612584 = r612572 <= r612583;
        double r612585 = r612580 * r612572;
        double r612586 = r612577 / r612585;
        double r612587 = r612577 / r612572;
        double r612588 = r612587 / r612580;
        double r612589 = r612584 ? r612586 : r612588;
        double r612590 = r612574 ? r612582 : r612589;
        return r612590;
}

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

Original6.8
Target2.2
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.93799498024277e-128

    1. Initial program 7.5

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

    2. Simplified6.2

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

    4. Applied associate-/r*2.5

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]
    5. Using strategy rm

    6. Applied div-inv2.5

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z} \cdot \frac{1}{y - t}}\]
    7. Using strategy rm

    8. Applied associate-*l/3.1

      \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot \frac{1}{y - t}}{z}}\]

    9. Simplified3.0

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

    if -6.93799498024277e-128 < z < 4.536023366252226e-37

    1. Initial program 3.5

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

    2. Simplified3.5

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

    4. Applied associate-/r*11.9

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]
    5. Using strategy rm

    6. Applied div-inv12.0

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z} \cdot \frac{1}{y - t}}\]
    7. Using strategy rm

    8. Applied frac-times3.5

      \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot 1}{z \cdot \left(y - t\right)}}\]

    9. Simplified3.5

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

    10. Simplified3.5

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

    if 4.536023366252226e-37 < z

    1. Initial program 9.5

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

    2. Simplified7.5

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

    4. Applied associate-/r*1.6

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.9379949802427703 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \mathbf{elif}\;z \le 4.5360233662522261 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))