Average Error: 16.0 → 8.4
Time: 25.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.0092107391768066 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, y + x\right)\\ \mathbf{elif}\;a \le 1.0196092676127147 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - t}{t - z}} + \left(y + x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.0092107391768066 \cdot 10^{-119}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, y + x\right)\\

\mathbf{elif}\;a \le 1.0196092676127147 \cdot 10^{-123}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{\frac{a - t}{t - z}} + \left(y + x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r25731652 = x;
        double r25731653 = y;
        double r25731654 = r25731652 + r25731653;
        double r25731655 = z;
        double r25731656 = t;
        double r25731657 = r25731655 - r25731656;
        double r25731658 = r25731657 * r25731653;
        double r25731659 = a;
        double r25731660 = r25731659 - r25731656;
        double r25731661 = r25731658 / r25731660;
        double r25731662 = r25731654 - r25731661;
        return r25731662;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25731663 = a;
        double r25731664 = -1.0092107391768066e-119;
        bool r25731665 = r25731663 <= r25731664;
        double r25731666 = y;
        double r25731667 = t;
        double r25731668 = r25731663 - r25731667;
        double r25731669 = r25731666 / r25731668;
        double r25731670 = z;
        double r25731671 = r25731667 - r25731670;
        double r25731672 = x;
        double r25731673 = r25731666 + r25731672;
        double r25731674 = fma(r25731669, r25731671, r25731673);
        double r25731675 = 1.0196092676127147e-123;
        bool r25731676 = r25731663 <= r25731675;
        double r25731677 = r25731670 / r25731667;
        double r25731678 = fma(r25731666, r25731677, r25731672);
        double r25731679 = r25731668 / r25731671;
        double r25731680 = r25731666 / r25731679;
        double r25731681 = r25731680 + r25731673;
        double r25731682 = r25731676 ? r25731678 : r25731681;
        double r25731683 = r25731665 ? r25731674 : r25731682;
        return r25731683;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.0
Target8.1
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.0092107391768066e-119

    1. Initial program 14.2

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

    2. Simplified7.5

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

    4. Applied div-inv7.5

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(t - z\right) \cdot \frac{1}{a - t}}, y + x\right)\]
    5. Using strategy rm

    6. Applied fma-udef7.5

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

    7. Simplified7.4

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

    9. Applied associate-/r/8.2

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

    10. Applied fma-def8.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, t - z, y + x\right)}\]

    if -1.0092107391768066e-119 < a < 1.0196092676127147e-123

    1. Initial program 19.8

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

    2. Simplified19.1

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

    4. Applied div-inv19.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(t - z\right) \cdot \frac{1}{a - t}}, y + x\right)\]

    5. Taylor expanded around inf 10.3

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

    6. Simplified9.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}\]

    if 1.0196092676127147e-123 < a

    1. Initial program 14.7

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

    2. Simplified8.2

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

    4. Applied div-inv8.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(t - z\right) \cdot \frac{1}{a - t}}, y + x\right)\]
    5. Using strategy rm

    6. Applied fma-udef8.2

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

    7. Simplified8.2

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.0092107391768066 \cdot 10^{-119}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, y + x\right)\\ \mathbf{elif}\;a \le 1.0196092676127147 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - t}{t - z}} + \left(y + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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