Average Error: 24.8 → 8.3
Time: 53.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - t \cdot \frac{1}{a - t}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\frac{x}{\frac{t}{z}} - \left(\frac{y}{\frac{t}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, y - x, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - t \cdot \frac{1}{a - t}, y - x, x\right)\\

\mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\
\;\;\;\;\frac{x}{\frac{t}{z}} - \left(\frac{y}{\frac{t}{z}} - y\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, y - x, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r29340671 = x;
        double r29340672 = y;
        double r29340673 = r29340672 - r29340671;
        double r29340674 = z;
        double r29340675 = t;
        double r29340676 = r29340674 - r29340675;
        double r29340677 = r29340673 * r29340676;
        double r29340678 = a;
        double r29340679 = r29340678 - r29340675;
        double r29340680 = r29340677 / r29340679;
        double r29340681 = r29340671 + r29340680;
        return r29340681;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r29340682 = x;
        double r29340683 = y;
        double r29340684 = r29340683 - r29340682;
        double r29340685 = z;
        double r29340686 = t;
        double r29340687 = r29340685 - r29340686;
        double r29340688 = r29340684 * r29340687;
        double r29340689 = a;
        double r29340690 = r29340689 - r29340686;
        double r29340691 = r29340688 / r29340690;
        double r29340692 = r29340682 + r29340691;
        double r29340693 = -1.5335432679341662e-301;
        bool r29340694 = r29340692 <= r29340693;
        double r29340695 = r29340685 / r29340690;
        double r29340696 = 1.0;
        double r29340697 = r29340696 / r29340690;
        double r29340698 = r29340686 * r29340697;
        double r29340699 = r29340695 - r29340698;
        double r29340700 = fma(r29340699, r29340684, r29340682);
        double r29340701 = 0.0;
        bool r29340702 = r29340692 <= r29340701;
        double r29340703 = r29340686 / r29340685;
        double r29340704 = r29340682 / r29340703;
        double r29340705 = r29340683 / r29340703;
        double r29340706 = r29340705 - r29340683;
        double r29340707 = r29340704 - r29340706;
        double r29340708 = r29340690 / r29340685;
        double r29340709 = r29340696 / r29340708;
        double r29340710 = r29340686 / r29340690;
        double r29340711 = r29340709 - r29340710;
        double r29340712 = fma(r29340711, r29340684, r29340682);
        double r29340713 = r29340702 ? r29340707 : r29340712;
        double r29340714 = r29340694 ? r29340700 : r29340713;
        return r29340714;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.8
Target9.3
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.5335432679341662e-301

    1. Initial program 21.4

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

    2. Simplified7.4

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

    4. Applied div-sub7.4

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

    6. Applied div-inv7.5

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

    8. Applied *-commutative7.5

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

    if -1.5335432679341662e-301 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 60.8

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

    2. Simplified60.8

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

    3. Taylor expanded around inf 17.6

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

    4. Simplified17.7

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

    if 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 21.6

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

    2. Simplified7.4

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

    4. Applied div-sub7.4

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

    6. Applied clear-num7.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{z}}} - \frac{t}{a - t}, y - x, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - t \cdot \frac{1}{a - t}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\frac{x}{\frac{t}{z}} - \left(\frac{y}{\frac{t}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{z}} - \frac{t}{a - t}, y - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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