Average Error: 0.1 → 0.0
Time: 1.7s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)
double f(double x, double y, double z) {
        double r1026200 = 4.0;
        double r1026201 = x;
        double r1026202 = y;
        double r1026203 = r1026201 - r1026202;
        double r1026204 = z;
        double r1026205 = 0.5;
        double r1026206 = r1026204 * r1026205;
        double r1026207 = r1026203 - r1026206;
        double r1026208 = r1026200 * r1026207;
        double r1026209 = r1026208 / r1026204;
        return r1026209;
}
↓
double f(double x, double y, double z) {
        double r1026210 = 4.0;
        double r1026211 = x;
        double r1026212 = z;
        double r1026213 = r1026211 / r1026212;
        double r1026214 = y;
        double r1026215 = r1026214 / r1026212;
        double r1026216 = r1026213 - r1026215;
        double r1026217 = 2.0;
        double r1026218 = -r1026217;
        double r1026219 = fma(r1026210, r1026216, r1026218);
        return r1026219;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.0
Herbie0.0
\[4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - \left(4 \cdot \frac{y}{z} + 2\right)}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)}\]
  4. Using strategy rm

  5. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z} - \frac{y}{z}}, -2\right)\]

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z))))

  (/ (* 4 (- (- x y) (* z 0.5))) z))