Average Error: 0.1 → 0.0
Time: 12.2s
Precision: 64
\[\frac{4.0 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{\frac{z}{4.0}}\]
\frac{4.0 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{\frac{z}{4.0}}
double f(double x, double y, double z) {
        double r15315761 = 4.0;
        double r15315762 = x;
        double r15315763 = y;
        double r15315764 = r15315762 - r15315763;
        double r15315765 = z;
        double r15315766 = 0.5;
        double r15315767 = r15315765 * r15315766;
        double r15315768 = r15315764 - r15315767;
        double r15315769 = r15315761 * r15315768;
        double r15315770 = r15315769 / r15315765;
        return r15315770;
}
↓
double f(double x, double y, double z) {
        double r15315771 = x;
        double r15315772 = z;
        double r15315773 = 0.5;
        double r15315774 = y;
        double r15315775 = fma(r15315772, r15315773, r15315774);
        double r15315776 = r15315771 - r15315775;
        double r15315777 = 4.0;
        double r15315778 = r15315772 / r15315777;
        double r15315779 = r15315776 / r15315778;
        return r15315779;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.0
Herbie0.0
\[4.0 \cdot \frac{x}{z} - \left(2.0 + 4.0 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4.0 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{\frac{z}{4.0}}}\]

  3. Final simplification0.0

    \[\leadsto \frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{\frac{z}{4.0}}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))