Average Error: 24.2 → 8.0
Time: 24.3s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -5.175543709426314317899237427278029417013 \cdot 10^{-293} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, x, \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\right)\\ \end{array}\]
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 -5.175543709426314317899237427278029417013 \cdot 10^{-293} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r414005 = x;
        double r414006 = y;
        double r414007 = r414006 - r414005;
        double r414008 = z;
        double r414009 = t;
        double r414010 = r414008 - r414009;
        double r414011 = r414007 * r414010;
        double r414012 = a;
        double r414013 = r414012 - r414009;
        double r414014 = r414011 / r414013;
        double r414015 = r414005 + r414014;
        return r414015;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r414016 = x;
        double r414017 = y;
        double r414018 = r414017 - r414016;
        double r414019 = z;
        double r414020 = t;
        double r414021 = r414019 - r414020;
        double r414022 = r414018 * r414021;
        double r414023 = a;
        double r414024 = r414023 - r414020;
        double r414025 = r414022 / r414024;
        double r414026 = r414016 + r414025;
        double r414027 = -5.175543709426314e-293;
        bool r414028 = r414026 <= r414027;
        double r414029 = 0.0;
        bool r414030 = r414026 <= r414029;
        double r414031 = !r414030;
        bool r414032 = r414028 || r414031;
        double r414033 = r414021 / r414024;
        double r414034 = fma(r414033, r414018, r414016);
        double r414035 = r414019 / r414020;
        double r414036 = -r414017;
        double r414037 = fma(r414035, r414036, r414017);
        double r414038 = fma(r414035, r414016, r414037);
        double r414039 = r414032 ? r414034 : r414038;
        return r414039;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.2
Target8.9
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -5.175543709426314e-293 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 21.0

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

    2. Simplified7.1

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

    if -5.175543709426314e-293 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 59.8

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

    2. Simplified59.8

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

    4. Applied div-inv59.8

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

    6. Applied add-cube-cbrt59.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\left(z - t\right) \cdot \frac{1}{a - t}} \cdot \sqrt[3]{\left(z - t\right) \cdot \frac{1}{a - t}}\right) \cdot \sqrt[3]{\left(z - t\right) \cdot \frac{1}{a - t}}}, y - x, x\right)\]

    7. Simplified59.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right)} \cdot \sqrt[3]{\left(z - t\right) \cdot \frac{1}{a - t}}, y - x, x\right)\]

    8. Simplified59.8

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

    9. Taylor expanded around inf 19.0

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

    10. Simplified19.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, x, \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -5.175543709426314317899237427278029417013 \cdot 10^{-293} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, x, \mathsf{fma}\left(\frac{z}{t}, -y, y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))