Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.9 → 1.1

Time: 19.4s

Precision: binary64

Cost: 2832

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

\[\frac{x \cdot y - z \cdot t}{a} \]

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := -1 \cdot \frac{z}{\frac{a}{t}} + x \cdot \frac{y}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+218}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t)))
        (t_2 (/ t_1 a))
        (t_3 (+ (* -1.0 (/ z (/ a t))) (* x (/ y a)))))
   (if (<= t_1 -5e+218)
     t_3
     (if (<= t_1 -1e-132)
       t_2
       (if (<= t_1 1e-123)
         t_3
         (if (<= t_1 2e+155) t_2 (+ (* z (- (/ t a))) (* y (/ x a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = t_1 / a;
	double t_3 = (-1.0 * (z / (a / t))) + (x * (y / a));
	double tmp;
	if (t_1 <= -5e+218) {
		tmp = t_3;
	} else if (t_1 <= -1e-132) {
		tmp = t_2;
	} else if (t_1 <= 1e-123) {
		tmp = t_3;
	} else if (t_1 <= 2e+155) {
		tmp = t_2;
	} else {
		tmp = (z * -(t / a)) + (y * (x / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - (z * t)) / a
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    t_2 = t_1 / a
    t_3 = ((-1.0d0) * (z / (a / t))) + (x * (y / a))
    if (t_1 <= (-5d+218)) then
        tmp = t_3
    else if (t_1 <= (-1d-132)) then
        tmp = t_2
    else if (t_1 <= 1d-123) then
        tmp = t_3
    else if (t_1 <= 2d+155) then
        tmp = t_2
    else
        tmp = (z * -(t / a)) + (y * (x / a))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = t_1 / a;
	double t_3 = (-1.0 * (z / (a / t))) + (x * (y / a));
	double tmp;
	if (t_1 <= -5e+218) {
		tmp = t_3;
	} else if (t_1 <= -1e-132) {
		tmp = t_2;
	} else if (t_1 <= 1e-123) {
		tmp = t_3;
	} else if (t_1 <= 2e+155) {
		tmp = t_2;
	} else {
		tmp = (z * -(t / a)) + (y * (x / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - (z * t)) / a

↓

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	t_2 = t_1 / a
	t_3 = (-1.0 * (z / (a / t))) + (x * (y / a))
	tmp = 0
	if t_1 <= -5e+218:
		tmp = t_3
	elif t_1 <= -1e-132:
		tmp = t_2
	elif t_1 <= 1e-123:
		tmp = t_3
	elif t_1 <= 2e+155:
		tmp = t_2
	else:
		tmp = (z * -(t / a)) + (y * (x / a))
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	t_2 = Float64(t_1 / a)
	t_3 = Float64(Float64(-1.0 * Float64(z / Float64(a / t))) + Float64(x * Float64(y / a)))
	tmp = 0.0
	if (t_1 <= -5e+218)
		tmp = t_3;
	elseif (t_1 <= -1e-132)
		tmp = t_2;
	elseif (t_1 <= 1e-123)
		tmp = t_3;
	elseif (t_1 <= 2e+155)
		tmp = t_2;
	else
		tmp = Float64(Float64(z * Float64(-Float64(t / a))) + Float64(y * Float64(x / a)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - (z * t)) / a;
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	t_2 = t_1 / a;
	t_3 = (-1.0 * (z / (a / t))) + (x * (y / a));
	tmp = 0.0;
	if (t_1 <= -5e+218)
		tmp = t_3;
	elseif (t_1 <= -1e-132)
		tmp = t_2;
	elseif (t_1 <= 1e-123)
		tmp = t_3;
	elseif (t_1 <= 2e+155)
		tmp = t_2;
	else
		tmp = (z * -(t / a)) + (y * (x / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 * N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+218], t$95$3, If[LessEqual[t$95$1, -1e-132], t$95$2, If[LessEqual[t$95$1, 1e-123], t$95$3, If[LessEqual[t$95$1, 2e+155], t$95$2, N[(N[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision] + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	7.9
Target	5.6
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.99999999999999983e218 or -9.9999999999999999e-133 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e-123

   1. Initial program 17.7

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 17.7

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]

   3. Simplified 2.3

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

      [Start] 17.7	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      rational.json-simplify-49 [=>] 10.3	\[ -1 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-49 [=>] 2.3	\[ -1 \cdot \left(z \cdot \frac{t}{a}\right) + \color{blue}{x \cdot \frac{y}{a}} \]

   4. Applied egg-rr 2.3

      \[\leadsto -1 \cdot \color{blue}{\frac{z}{\frac{a}{t}}} + x \cdot \frac{y}{a} \]

   if -4.99999999999999983e218 < (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999999e-133 or 1.0000000000000001e-123 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000001e155

   1. Initial program 0.3

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 2.00000000000000001e155 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 21.1

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 21.1

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]

   3. Simplified 2.6

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

      [Start] 21.1	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      rational.json-simplify-49 [=>] 12.0	\[ -1 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-43 [=>] 12.0	\[ \color{blue}{z \cdot \left(\frac{t}{a} \cdot -1\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-9 [=>] 12.0	\[ z \cdot \color{blue}{\left(-\frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-2 [=>] 12.0	\[ z \cdot \left(-\frac{t}{a}\right) + \frac{\color{blue}{x \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 2.6	\[ z \cdot \left(-\frac{t}{a}\right) + \color{blue}{y \cdot \frac{x}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+218}:\\ \;\;\;\;-1 \cdot \frac{z}{\frac{a}{t}} + x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{-132}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{-123}:\\ \;\;\;\;-1 \cdot \frac{z}{\frac{a}{t}} + x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	2832

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+293}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	0.9
Cost	2832

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := -1 \cdot \left(z \cdot \frac{t}{a}\right) + x \cdot \frac{y}{a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+293}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 3
Error	4.8
Cost	1616

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+273}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-314}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 4
Error	26.0
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	26.0
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 6
Error	26.8
Cost	1176

\[\begin{array}{l} t_1 := \frac{t \cdot \left(-z\right)}{a}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 7
Error	26.9
Cost	1176

\[\begin{array}{l} t_1 := \frac{t \cdot \left(-z\right)}{a}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{\frac{1}{x}}}{a}\\ \mathbf{elif}\;z \leq -1.42 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 8
Error	32.7
Cost	584

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+123}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	32.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+121}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10
Error	31.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 11
Error	32.6
Cost	320

\[x \cdot \frac{y}{a} \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))