Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Average Error: 3.3 → 0.4

Time: 2.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+132)
   (* x (- (fma y z 1.0) z))
   (if (<= x 4e+51) (fma z (- (* x y) x) x) (fma (* x z) (+ y -1.0) x))))

C

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 tmp;
	if (x <= -1e+132) {
		tmp = x * (fma(y, z, 1.0) - z);
	} else if (x <= 4e+51) {
		tmp = fma(z, ((x * y) - x), x);
	} else {
		tmp = fma((x * z), (y + -1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+132)
		tmp = Float64(x * Float64(fma(y, z, 1.0) - z));
	elseif (x <= 4e+51)
		tmp = fma(z, Float64(Float64(x * y) - x), x);
	else
		tmp = fma(Float64(x * z), Float64(y + -1.0), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[x, -1e+132], N[(x * N[(N[(y * z + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e+51], N[(z * N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision] + x), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.3
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999991e131

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 0.1

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

   3. Simplified 0.1

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

   4. Taylor expanded in z around 0 0.1

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

   5. Simplified 0.1

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

   if -9.99999999999999991e131 < x < 4e51

   1. Initial program 4.3

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

   2. Simplified 0.6

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

   if 4e51 < x

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot x, y + -1, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))