Average Error: 10.5 → 0.0
Time: 14.6s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right)
double f(double x, double y, double z) {
        double r539043 = x;
        double r539044 = y;
        double r539045 = z;
        double r539046 = r539045 - r539043;
        double r539047 = r539044 * r539046;
        double r539048 = r539043 + r539047;
        double r539049 = r539048 / r539045;
        return r539049;
}
↓
double f(double x, double y, double z) {
        double r539050 = x;
        double r539051 = z;
        double r539052 = r539050 / r539051;
        double r539053 = 1.0;
        double r539054 = y;
        double r539055 = r539053 - r539054;
        double r539056 = fma(r539052, r539055, r539054);
        return r539056;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.5

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

  2. Taylor expanded around 0 3.9

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

  3. Simplified3.9

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

  5. Applied *-un-lft-identity3.9

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

  6. Applied times-frac0.0

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

  7. Applied *-un-lft-identity0.0

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

  8. Applied distribute-rgt-out--0.0

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

  9. Applied fma-def0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

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

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