Average Error: 24.6 → 10.3
Time: 53.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -7.9831762004291433393554174793802804664 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\ \mathbf{elif}\;a \le 1.742406464651130897468754713538689990604 \cdot 10^{-205}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z}{t} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -7.9831762004291433393554174793802804664 \cdot 10^{-139}:\\
\;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\

\mathbf{elif}\;a \le 1.742406464651130897468754713538689990604 \cdot 10^{-205}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z}{t} \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r32855088 = x;
        double r32855089 = y;
        double r32855090 = r32855089 - r32855088;
        double r32855091 = z;
        double r32855092 = t;
        double r32855093 = r32855091 - r32855092;
        double r32855094 = r32855090 * r32855093;
        double r32855095 = a;
        double r32855096 = r32855095 - r32855092;
        double r32855097 = r32855094 / r32855096;
        double r32855098 = r32855088 + r32855097;
        return r32855098;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r32855099 = a;
        double r32855100 = -7.983176200429143e-139;
        bool r32855101 = r32855099 <= r32855100;
        double r32855102 = y;
        double r32855103 = x;
        double r32855104 = r32855102 - r32855103;
        double r32855105 = cbrt(r32855104);
        double r32855106 = z;
        double r32855107 = t;
        double r32855108 = r32855106 - r32855107;
        double r32855109 = r32855099 - r32855107;
        double r32855110 = r32855108 / r32855109;
        double r32855111 = cbrt(r32855110);
        double r32855112 = r32855105 * r32855111;
        double r32855113 = r32855104 * r32855110;
        double r32855114 = cbrt(r32855113);
        double r32855115 = r32855112 * r32855114;
        double r32855116 = fma(r32855115, r32855114, r32855103);
        double r32855117 = 1.742406464651131e-205;
        bool r32855118 = r32855099 <= r32855117;
        double r32855119 = r32855103 / r32855107;
        double r32855120 = r32855106 / r32855107;
        double r32855121 = r32855120 * r32855102;
        double r32855122 = r32855102 - r32855121;
        double r32855123 = fma(r32855119, r32855106, r32855122);
        double r32855124 = r32855118 ? r32855123 : r32855116;
        double r32855125 = r32855101 ? r32855116 : r32855124;
        return r32855125;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.6
Target9.4
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -7.983176200429143e-139 or 1.742406464651131e-205 < a

    1. Initial program 23.4

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

    2. Simplified9.7

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

    4. Applied fma-udef9.7

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

    6. Applied add-cube-cbrt10.2

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

    8. Applied cbrt-prod10.2

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

    10. Applied fma-def10.2

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

    if -7.983176200429143e-139 < a < 1.742406464651131e-205

    1. Initial program 30.1

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

    2. Simplified20.1

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

    4. Applied fma-udef20.1

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

    5. Taylor expanded around inf 13.8

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

    6. Simplified10.6

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.9831762004291433393554174793802804664 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\ \mathbf{elif}\;a \le 1.742406464651130897468754713538689990604 \cdot 10^{-205}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z}{t} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, \sqrt[3]{\left(y - x\right) \cdot \frac{z - t}{a - t}}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))))