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

Percentage Accurate: 91.2% → 97.4%

Time: 10.5s

Alternatives: 10

Speedup: 0.7×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, x, \frac{z}{a} \cdot \left(-t\right)\right)\\ t_2 := y \cdot x - t \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) x (* (/ z a) (- t)))) (t_2 (- (* y x) (* t z))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 5e+231) (/ (fma y x (* (- z) t)) a) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), x, ((z / a) * -t));
	double t_2 = (y * x) - (t * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+231) {
		tmp = fma(y, x, (-z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), x, Float64(Float64(z / a) * Float64(-t)))
	t_2 = Float64(Float64(y * x) - Float64(t * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+231)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * x + N[(N[(z / a), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+231], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 5.00000000000000028e231 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 75.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, x, \mathsf{neg}\left(\frac{z \cdot t}{a}\right)\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, x, \mathsf{neg}\left(\frac{z \cdot t}{a}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \mathsf{neg}\left(\frac{\color{blue}{z \cdot t}}{a}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \mathsf{neg}\left(\frac{\color{blue}{t \cdot z}}{a}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \frac{z}{a}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \frac{z}{a}}\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot \frac{z}{a}\right) \]

      16. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000028e231

   1. Initial program 99.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(z \cdot t\right)\right)}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{z \cdot t}\right)\right)}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{t \cdot z}\right)\right)}{a} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot z}\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot z}\right)}{a} \]

      10. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot z \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{z}{a} \cdot \left(-t\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot z \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{z}{a} \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot x - t \cdot z}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- (* y x) (* t z)) a) (- INFINITY)) (* (/ y a) x) (/ (* y x) a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((y * x) - (t * z)) / a) <= -((double) INFINITY)) {
		tmp = (y / a) * x;
	} else {
		tmp = (y * x) / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((y * x) - (t * z)) / a) <= -Double.POSITIVE_INFINITY) {
		tmp = (y / a) * x;
	} else {
		tmp = (y * x) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (((y * x) - (t * z)) / a) <= -math.inf:
		tmp = (y / a) * x
	else:
		tmp = (y * x) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) - Float64(t * z)) / a) <= Float64(-Inf))
		tmp = Float64(Float64(y / a) * x);
	else
		tmp = Float64(Float64(y * x) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((((y * x) - (t * z)) / a) <= -Inf)
		tmp = (y / a) * x;
	else
		tmp = (y * x) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], (-Infinity)], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0

   1. Initial program 80.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

      2. lower-*.f64 45.5

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

   5. Applied rewrites 45.5%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]

      4. lower-/.f64 57.1

         \[\leadsto \color{blue}{\frac{y}{a}} \cdot x \]

   8. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

      2. lower-*.f64 51.8

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]

   5. Applied rewrites 51.8%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x - t \cdot z}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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