Average Error: 10.6 → 1.4
Time: 47.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r25781101 = x;
        double r25781102 = y;
        double r25781103 = z;
        double r25781104 = t;
        double r25781105 = r25781103 - r25781104;
        double r25781106 = r25781102 * r25781105;
        double r25781107 = a;
        double r25781108 = r25781107 - r25781104;
        double r25781109 = r25781106 / r25781108;
        double r25781110 = r25781101 + r25781109;
        return r25781110;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25781111 = z;
        double r25781112 = t;
        double r25781113 = r25781111 - r25781112;
        double r25781114 = a;
        double r25781115 = r25781114 - r25781112;
        double r25781116 = r25781113 / r25781115;
        double r25781117 = y;
        double r25781118 = x;
        double r25781119 = fma(r25781116, r25781117, r25781118);
        return r25781119;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.6
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.6

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

  2. Simplified2.8

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

  4. Applied div-inv2.9

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

  6. Applied fma-udef2.9

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

  7. Simplified10.6

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

  9. Applied *-un-lft-identity10.6

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

  10. Applied *-un-lft-identity10.6

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

  11. Applied distribute-lft-out10.6

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

  12. Simplified1.4

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

  13. Final simplification1.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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