Average Error: 14.7 → 7.6
Time: 6.2s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot t + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot t + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r93563 = x;
        double r93564 = y;
        double r93565 = z;
        double r93566 = r93564 - r93565;
        double r93567 = t;
        double r93568 = r93567 - r93563;
        double r93569 = a;
        double r93570 = r93569 - r93565;
        double r93571 = r93568 / r93570;
        double r93572 = r93566 * r93571;
        double r93573 = r93563 + r93572;
        return r93573;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r93574 = y;
        double r93575 = z;
        double r93576 = r93574 - r93575;
        double r93577 = 1.0;
        double r93578 = a;
        double r93579 = r93578 - r93575;
        double r93580 = r93577 / r93579;
        double r93581 = r93576 * r93580;
        double r93582 = t;
        double r93583 = r93581 * r93582;
        double r93584 = x;
        double r93585 = -r93584;
        double r93586 = r93576 / r93579;
        double r93587 = fma(r93585, r93586, r93584);
        double r93588 = r93583 + r93587;
        return r93588;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 14.7

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

  2. Simplified14.6

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

  4. Applied div-sub14.6

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

  6. Applied div-inv14.6

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

  8. Applied fma-udef14.7

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

  9. Simplified11.6

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

  11. Applied sub-neg11.6

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

  12. Applied distribute-lft-in11.6

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

  13. Applied associate-+l+8.3

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

  14. Simplified7.6

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

  15. Final simplification7.6

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))