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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r546503 = x;
        double r546504 = y;
        double r546505 = r546504 - r546503;
        double r546506 = z;
        double r546507 = t;
        double r546508 = r546506 / r546507;
        double r546509 = r546505 * r546508;
        double r546510 = r546503 + r546509;
        return r546510;
}

↓

double f(double x, double y, double z, double t) {
        double r546511 = y;
        double r546512 = x;
        double r546513 = r546511 - r546512;
        double r546514 = z;
        double r546515 = t;
        double r546516 = r546514 / r546515;
        double r546517 = fma(r546513, r546516, r546512);
        return r546517;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	2.1
Target	2.2
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

Initial program 2.1
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified2.1
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification2.1
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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