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

Average Error: 3.4 → 0.4

Time: 8.0s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= t_0 -5e+114)
     (* (* z x) (+ y -1.0))
     (if (<= t_0 5e+302) (+ x (* x (* z (+ y -1.0)))) (* y (* z 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 t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -5e+114) {
		tmp = (z * x) * (y + -1.0);
	} else if (t_0 <= 5e+302) {
		tmp = x + (x * (z * (y + -1.0)));
	} else {
		tmp = y * (z * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - ((1.0d0 - y) * z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - y) * z
    if (t_0 <= (-5d+114)) then
        tmp = (z * x) * (y + (-1.0d0))
    else if (t_0 <= 5d+302) then
        tmp = x + (x * (z * (y + (-1.0d0))))
    else
        tmp = y * (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -5e+114) {
		tmp = (z * x) * (y + -1.0);
	} else if (t_0 <= 5e+302) {
		tmp = x + (x * (z * (y + -1.0)));
	} else {
		tmp = y * (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

↓

def code(x, y, z):
	t_0 = (1.0 - y) * z
	tmp = 0
	if t_0 <= -5e+114:
		tmp = (z * x) * (y + -1.0)
	elif t_0 <= 5e+302:
		tmp = x + (x * (z * (y + -1.0)))
	else:
		tmp = y * (z * 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)
	t_0 = Float64(Float64(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= -5e+114)
		tmp = Float64(Float64(z * x) * Float64(y + -1.0));
	elseif (t_0 <= 5e+302)
		tmp = Float64(x + Float64(x * Float64(z * Float64(y + -1.0))));
	else
		tmp = Float64(y * Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	tmp = 0.0;
	if (t_0 <= -5e+114)
		tmp = (z * x) * (y + -1.0);
	elseif (t_0 <= 5e+302)
		tmp = x + (x * (z * (y + -1.0)));
	else
		tmp = y * (z * x);
	end
	tmp_2 = 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_] := Block[{t$95$0 = N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+114], N[(N[(z * x), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(x + N[(x * N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.4
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 (*.f64 (-.f64 1 y) z) < -5.0000000000000001e114

   1. Initial program 10.9

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

   2. Simplified 10.9

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

      [Start] 10.9	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      cancel-sign-sub-inv [=>] 10.9	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right)\right) \cdot z\right)} \]
      +-commutative [=>] 10.9	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z + 1\right)} \]
      *-commutative [=>] 10.9	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 10.9	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]
      sub-neg [=>] 10.9	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(1 + \left(-y\right)\right)}, 1\right) \]
      +-commutative [=>] 10.9	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(\left(-y\right) + 1\right)}, 1\right) \]
      distribute-neg-in [=>] 10.9	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{\left(-\left(-y\right)\right) + \left(-1\right)}, 1\right) \]
      remove-double-neg [=>] 10.9	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{y} + \left(-1\right), 1\right) \]
      metadata-eval [=>] 10.9	\[ x \cdot \mathsf{fma}\left(z, y + \color{blue}{-1}, 1\right) \]

   3. Taylor expanded in z around inf 1.3

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

   4. Simplified 1.5

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

      [Start] 1.3	\[ z \cdot \left(\left(y - 1\right) \cdot x\right) \]
      *-commutative [=>] 1.3	\[ z \cdot \color{blue}{\left(x \cdot \left(y - 1\right)\right)} \]
      sub-neg [=>] 1.3	\[ z \cdot \left(x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) \]
      metadata-eval [=>] 1.3	\[ z \cdot \left(x \cdot \left(y + \color{blue}{-1}\right)\right) \]
      associate-*r* [=>] 1.5	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(y + -1\right)} \]

   if -5.0000000000000001e114 < (*.f64 (-.f64 1 y) z) < 5e302

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 5e302 < (*.f64 (-.f64 1 y) z)

   1. Initial program 57.1

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

   2. Taylor expanded in y around inf 3.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x + x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 2
Error	12.9
Cost	1113

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+36} \lor \neg \left(y \leq 7.5 \cdot 10^{+56}\right) \land y \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	13.0
Cost	1113

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+36}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+55} \lor \neg \left(y \leq 8.2 \cdot 10^{+111}\right):\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.2
Cost	1048

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ t_1 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.2
Cost	1048

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+178}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+73}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -120 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 7
Error	19.7
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	33.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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))))