Average Error: 3.4 → 3.4
Time: 16.1s
Precision: 64
\[x \cdot \left(1.0 - \left(1.0 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(z, y - 1.0, 1.0\right) \cdot x\]
x \cdot \left(1.0 - \left(1.0 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(z, y - 1.0, 1.0\right) \cdot x
double f(double x, double y, double z) {
        double r36652095 = x;
        double r36652096 = 1.0;
        double r36652097 = y;
        double r36652098 = r36652096 - r36652097;
        double r36652099 = z;
        double r36652100 = r36652098 * r36652099;
        double r36652101 = r36652096 - r36652100;
        double r36652102 = r36652095 * r36652101;
        return r36652102;
}
↓
double f(double x, double y, double z) {
        double r36652103 = z;
        double r36652104 = y;
        double r36652105 = 1.0;
        double r36652106 = r36652104 - r36652105;
        double r36652107 = fma(r36652103, r36652106, r36652105);
        double r36652108 = x;
        double r36652109 = r36652107 * r36652108;
        return r36652109;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1.0 - \left(1.0 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1.0 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1.0 - \left(1.0 - y\right) \cdot z\right) \lt 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1.0 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

    \[x \cdot \left(1.0 - \left(1.0 - y\right) \cdot z\right)\]

  2. Simplified3.4

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(z, y - 1.0, 1.0\right)}\]

  3. Final simplification3.4

    \[\leadsto \mathsf{fma}\left(z, y - 1.0, 1.0\right) \cdot x\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))