Average Error: 24.1 → 7.3
Time: 5.2s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[y \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + \mathsf{fma}\left(-x, \frac{z - t}{a - t}, x\right)\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
y \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + \mathsf{fma}\left(-x, \frac{z - t}{a - t}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r504150 = x;
        double r504151 = y;
        double r504152 = r504151 - r504150;
        double r504153 = z;
        double r504154 = t;
        double r504155 = r504153 - r504154;
        double r504156 = r504152 * r504155;
        double r504157 = a;
        double r504158 = r504157 - r504154;
        double r504159 = r504156 / r504158;
        double r504160 = r504150 + r504159;
        return r504160;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r504161 = y;
        double r504162 = z;
        double r504163 = t;
        double r504164 = r504162 - r504163;
        double r504165 = 1.0;
        double r504166 = a;
        double r504167 = r504166 - r504163;
        double r504168 = r504165 / r504167;
        double r504169 = r504164 * r504168;
        double r504170 = r504161 * r504169;
        double r504171 = x;
        double r504172 = -r504171;
        double r504173 = r504164 / r504167;
        double r504174 = fma(r504172, r504173, r504171);
        double r504175 = r504170 + r504174;
        return r504175;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.1
Target9.0
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 24.1

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

  2. Simplified14.6

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

  4. Applied div-inv14.7

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

  6. Applied add-cube-cbrt15.3

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

  7. Applied associate-*r*15.3

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

  9. Applied fma-udef15.3

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

  10. Simplified11.5

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

  12. Applied sub-neg11.5

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

  13. Applied distribute-rgt-in11.5

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

  14. Applied associate-+l+8.0

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

  15. Simplified7.3

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

  16. Final simplification7.3

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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