Average Error: 3.3 → 0.1
Time: 2.8s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\begin{array}{l} t_0 := x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ t_1 := \mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 (* z (+ y -1.0))))) (t_1 (fma (* x z) (+ y -1.0) x)))
   (if (<= t_0 -4e+49) t_1 (if (<= t_0 2e+47) (fma z (- (* x y) x) x) t_1))))
double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}
↓
double code(double x, double y, double z) {
	double t_0 = x * (1.0 + (z * (y + -1.0)));
	double t_1 = fma((x * z), (y + -1.0), x);
	double tmp;
	if (t_0 <= -4e+49) {
		tmp = t_1;
	} else if (t_0 <= 2e+47) {
		tmp = fma(z, ((x * y) - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end
↓
function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))))
	t_1 = fma(Float64(x * z), Float64(y + -1.0), x)
	tmp = 0.0
	if (t_0 <= -4e+49)
		tmp = t_1;
	elseif (t_0 <= 2e+47)
		tmp = fma(z, Float64(Float64(x * y) - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+49], t$95$1, If[LessEqual[t$95$0, 2e+47], N[(z * N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\begin{array}{l}
t_0 := x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\
t_1 := \mathsf{fma}\left(x \cdot z, y + -1, x\right)\\
\mathbf{if}\;t_0 \leq -4 \cdot 10^{+49}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_0 \leq 2 \cdot 10^{+47}:\\
\;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.3
Target0.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -3.99999999999999979e49 or 2.0000000000000001e47 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

    1. Initial program 6.4

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

    2. Taylor expanded in x around 0 6.4

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

    3. Simplified0.1

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

    if -3.99999999999999979e49 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 2.0000000000000001e47

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x \cdot y - x, x\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right) \leq -4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \mathbf{elif}\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right) \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :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))))