Average Error: 11.3 → 1.2
Time: 4.8s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{1}{\frac{a - z}{y - z}}, t, x\right)\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\mathsf{fma}\left(1 \cdot \frac{1}{\frac{a - z}{y - z}}, t, x\right)
double f(double x, double y, double z, double t, double a) {
        double r648559 = x;
        double r648560 = y;
        double r648561 = z;
        double r648562 = r648560 - r648561;
        double r648563 = t;
        double r648564 = r648562 * r648563;
        double r648565 = a;
        double r648566 = r648565 - r648561;
        double r648567 = r648564 / r648566;
        double r648568 = r648559 + r648567;
        return r648568;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r648569 = 1.0;
        double r648570 = a;
        double r648571 = z;
        double r648572 = r648570 - r648571;
        double r648573 = y;
        double r648574 = r648573 - r648571;
        double r648575 = r648572 / r648574;
        double r648576 = r648569 / r648575;
        double r648577 = r648569 * r648576;
        double r648578 = t;
        double r648579 = x;
        double r648580 = fma(r648577, r648578, r648579);
        return r648580;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.3
Target0.5
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified1.2

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

  4. Applied *-un-lft-identity1.2

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

  5. Applied *-un-lft-identity1.2

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

  6. Applied times-frac1.2

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

  7. Simplified1.2

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

  9. Applied clear-num1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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