Average Error: 24.2 → 8.5
Time: 18.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\frac{z}{a - t} \cdot \left(y - x\right) + \mathsf{fma}\left(-\frac{t}{a - t}, y - x, x\right)\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\frac{z}{a - t} \cdot \left(y - x\right) + \mathsf{fma}\left(-\frac{t}{a - t}, y - x, x\right)
double f(double x, double y, double z, double t, double a) {
        double r631112 = x;
        double r631113 = y;
        double r631114 = r631113 - r631112;
        double r631115 = z;
        double r631116 = t;
        double r631117 = r631115 - r631116;
        double r631118 = r631114 * r631117;
        double r631119 = a;
        double r631120 = r631119 - r631116;
        double r631121 = r631118 / r631120;
        double r631122 = r631112 + r631121;
        return r631122;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r631123 = z;
        double r631124 = a;
        double r631125 = t;
        double r631126 = r631124 - r631125;
        double r631127 = r631123 / r631126;
        double r631128 = y;
        double r631129 = x;
        double r631130 = r631128 - r631129;
        double r631131 = r631127 * r631130;
        double r631132 = r631125 / r631126;
        double r631133 = -r631132;
        double r631134 = fma(r631133, r631130, r631129);
        double r631135 = r631131 + r631134;
        return r631135;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.2
Target9.3
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 24.2

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

  2. Simplified14.6

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

  4. Applied fma-udef14.7

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

  6. Applied div-inv14.7

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

  7. Applied associate-*l*11.7

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

  8. Simplified11.6

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

  10. Applied div-sub11.6

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

  12. Applied sub-neg11.6

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

  13. Applied distribute-rgt-in11.6

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

  14. Applied associate-+l+8.6

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

  15. Simplified8.5

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

  16. Final simplification8.5

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

Reproduce

herbie shell --seed 2020045 +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))))