Average Error: 3.4 → 1.4
Time: 2.9s
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(x, 1, \left(x \cdot z\right) \cdot \left(y - 1\right)\right)\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(x, 1, \left(x \cdot z\right) \cdot \left(y - 1\right)\right)
double code(double x, double y, double z) {
	return (x * (1.0 - ((1.0 - y) * z)));
}
↓
double code(double x, double y, double z) {
	return fma(x, 1.0, ((x * z) * (y - 1.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.4
Target0.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

    \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]

  2. Simplified3.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - 1, z, 1\right) \cdot x}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt4.7

    \[\leadsto \mathsf{fma}\left(y - 1, z, 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\]

  5. Applied associate-*r*4.7

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(y - 1, z, 1\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}\]

  6. Taylor expanded around inf 3.4

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

  7. Simplified1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1 (* (- 1 y) z))) -1.618195973607049e+50) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x)))))

  (* x (- 1 (* (- 1 y) z))))