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

Percentage Accurate: 91.2% → 95.0%

Time: 6.3s

Alternatives: 8

Speedup: 0.5×

• 28.1% 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: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+257}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \end{array} \]

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 (<= (* z t) -5e+307)
   (/ z (/ (- a) t))
   (if (<= (* z t) 1e+257) (/ (- (* x y) (* z t)) a) (* t (/ (- z) 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 ((z * t) <= -5e+307) {
		tmp = z / (-a / t);
	} else if ((z * t) <= 1e+257) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t * (-z / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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) :: tmp
    if ((z * t) <= (-5d+307)) then
        tmp = z / (-a / t)
    else if ((z * t) <= 1d+257) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t * (-z / a)
    end if
    code = tmp
end function

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 ((z * t) <= -5e+307) {
		tmp = z / (-a / t);
	} else if ((z * t) <= 1e+257) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t * (-z / 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 (z * t) <= -5e+307:
		tmp = z / (-a / t)
	elif (z * t) <= 1e+257:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t * (-z / 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(z * t) <= -5e+307)
		tmp = Float64(z / Float64(Float64(-a) / t));
	elseif (Float64(z * t) <= 1e+257)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(t * Float64(Float64(-z) / 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 ((z * t) <= -5e+307)
		tmp = z / (-a / t);
	elseif ((z * t) <= 1e+257)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t * (-z / 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[(z * t), $MachinePrecision], -5e+307], N[(z / N[((-a) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+257], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e307

   1. Initial program 69.7%

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

   2. Taylor expanded in x around 0 80.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 80.2%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. distribute-frac-neg 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 80.2%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. distribute-neg-frac 99.9%

         \[\leadsto \color{blue}{\frac{-\left(-z\right)}{-\frac{a}{t}}} \]

      6. remove-double-neg 99.9%

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

      7. distribute-neg-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -5e307 < (*.f64 z t) < 1.00000000000000003e257

   1. Initial program 95.2%

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

   if 1.00000000000000003e257 < (*.f64 z t)

   1. Initial program 59.9%

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

   2. Taylor expanded in x around 0 59.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 59.9%

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

      2. associate-*l/ 94.7%

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

      3. associate-*r* 94.7%

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

      4. neg-mul-1 94.7%

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

      5. distribute-frac-neg 94.7%

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

   4. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+257}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 2: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \end{array} \]

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 (<= (* z t) 1e+257) (/ (fma x y (* z (- t))) a) (* t (/ (- z) 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 ((z * t) <= 1e+257) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = t * (-z / 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(z * t) <= 1e+257)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(t * Float64(Float64(-z) / a));
	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_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+257], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.00000000000000003e257

   1. Initial program 93.1%

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

      1. div-sub 91.0%

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

      2. *-commutative 91.0%

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

      3. div-sub 93.1%

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

      4. fma-neg 94.0%

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

      5. *-commutative 94.0%

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

      6. distribute-rgt-neg-out 94.0%

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

   3. Simplified 94.0%

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

   if 1.00000000000000003e257 < (*.f64 z t)

   1. Initial program 59.9%

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

   2. Taylor expanded in x around 0 59.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 59.9%

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

      2. associate-*l/ 94.7%

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

      3. associate-*r* 94.7%

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

      4. neg-mul-1 94.7%

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

      5. distribute-frac-neg 94.7%

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

   4. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+257}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 3: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.2:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 40000:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

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 (<= (* z t) -0.2)
   (* t (/ (- z) a))
   (if (<= (* z t) 5e-190)
     (/ (* x y) a)
     (if (<= (* z t) 40000.0) (* x (/ y a)) (/ (- t) (/ a z))))))

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 ((z * t) <= -0.2) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 5e-190) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 40000.0) {
		tmp = x * (y / a);
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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) :: tmp
    if ((z * t) <= (-0.2d0)) then
        tmp = t * (-z / a)
    else if ((z * t) <= 5d-190) then
        tmp = (x * y) / a
    else if ((z * t) <= 40000.0d0) then
        tmp = x * (y / a)
    else
        tmp = -t / (a / z)
    end if
    code = tmp
end function

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 ((z * t) <= -0.2) {
		tmp = t * (-z / a);
	} else if ((z * t) <= 5e-190) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 40000.0) {
		tmp = x * (y / a);
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -0.2:
		tmp = t * (-z / a)
	elif (z * t) <= 5e-190:
		tmp = (x * y) / a
	elif (z * t) <= 40000.0:
		tmp = x * (y / a)
	else:
		tmp = -t / (a / z)
	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(z * t) <= -0.2)
		tmp = Float64(t * Float64(Float64(-z) / a));
	elseif (Float64(z * t) <= 5e-190)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(z * t) <= 40000.0)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	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 ((z * t) <= -0.2)
		tmp = t * (-z / a);
	elseif ((z * t) <= 5e-190)
		tmp = (x * y) / a;
	elseif ((z * t) <= 40000.0)
		tmp = x * (y / a);
	else
		tmp = -t / (a / z);
	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[(z * t), $MachinePrecision], -0.2], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-190], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 40000.0], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -0.20000000000000001

   1. Initial program 89.5%

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

   2. Taylor expanded in x around 0 73.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 73.2%

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

      2. associate-*l/ 74.9%

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

      3. associate-*r* 74.9%

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

      4. neg-mul-1 74.9%

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

      5. distribute-frac-neg 74.9%

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

   4. Simplified 74.9%

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

   if -0.20000000000000001 < (*.f64 z t) < 5.00000000000000034e-190

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 86.8%

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

   if 5.00000000000000034e-190 < (*.f64 z t) < 4e4

   1. Initial program 84.0%

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

   2. Taylor expanded in x around inf 59.5%

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

      1. associate-*l/ 68.3%

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

   4. Simplified 68.3%

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

      1. associate-*l/ 59.5%

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

      2. clear-num 59.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y}}} \]

   6. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-/r* 67.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{x}}{y}}} \]

      2. clear-num 67.4%

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

      3. associate-/r/ 69.2%

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

   8. Applied egg-rr 69.2%

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

   if 4e4 < (*.f64 z t)

   1. Initial program 86.2%

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

   2. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 67.9%

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

      2. associate-/l* 77.1%

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

   4. Simplified 77.1%

      \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.2:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 40000:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 4: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.0008:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

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 (<= (* x y) -0.0008)
   (/ (* x y) a)
   (if (<= (* x y) 5e+45) (/ (- t) (/ a z)) (/ x (/ a y)))))

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 ((x * y) <= -0.0008) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 5e+45) {
		tmp = -t / (a / z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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) :: tmp
    if ((x * y) <= (-0.0008d0)) then
        tmp = (x * y) / a
    else if ((x * y) <= 5d+45) then
        tmp = -t / (a / z)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

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 ((x * y) <= -0.0008) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 5e+45) {
		tmp = -t / (a / z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -0.0008:
		tmp = (x * y) / a
	elif (x * y) <= 5e+45:
		tmp = -t / (a / z)
	else:
		tmp = x / (a / y)
	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(x * y) <= -0.0008)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 5e+45)
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(x / Float64(a / y));
	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 ((x * y) <= -0.0008)
		tmp = (x * y) / a;
	elseif ((x * y) <= 5e+45)
		tmp = -t / (a / z);
	else
		tmp = x / (a / y);
	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[(x * y), $MachinePrecision], -0.0008], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+45], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -8.00000000000000038e-4

   1. Initial program 89.3%

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

   2. Taylor expanded in x around inf 76.1%

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

   if -8.00000000000000038e-4 < (*.f64 x y) < 5e45

   1. Initial program 92.0%

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

   2. Taylor expanded in x around 0 71.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 71.5%

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

      2. associate-/l* 72.7%

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

   4. Simplified 72.7%

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

   if 5e45 < (*.f64 x y)

   1. Initial program 88.7%

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

   2. Taylor expanded in x around inf 77.7%

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

      1. associate-*l/ 81.3%

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

   4. Simplified 81.3%

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

      1. associate-/r/ 78.1%

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

   6. Applied egg-rr 78.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.0008:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

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 (<= (* x y) -2e-159) (/ (* x y) a) (/ x (/ a y))))

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 ((x * y) <= -2e-159) {
		tmp = (x * y) / a;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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) :: tmp
    if ((x * y) <= (-2d-159)) then
        tmp = (x * y) / a
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

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 ((x * y) <= -2e-159) {
		tmp = (x * y) / a;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-159:
		tmp = (x * y) / a
	else:
		tmp = x / (a / y)
	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(x * y) <= -2e-159)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(x / Float64(a / y));
	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 ((x * y) <= -2e-159)
		tmp = (x * y) / a;
	else
		tmp = x / (a / y);
	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[(x * y), $MachinePrecision], -2e-159], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999998e-159

   1. Initial program 92.4%

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

   2. Taylor expanded in x around inf 66.6%

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

   if -1.99999999999999998e-159 < (*.f64 x y)

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 46.5%

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

      1. associate-*l/ 46.3%

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

   4. Simplified 46.3%

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

      1. associate-/r/ 47.0%

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

   6. Applied egg-rr 47.0%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ y \cdot \frac{x}{a} \end{array} \]

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 (* 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) {
	return y * (x / a);
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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
    code = y * (x / a)
end function

Java

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

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return y * (x / a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(y * Float64(x / a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = y * (x / a);
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_] := N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in x around inf 53.7%

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

   1. associate-*l/ 51.1%

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

4. Simplified 51.1%

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

5. Final simplification 51.1%

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

Alternative 7: 50.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ x \cdot \frac{y}{a} \end{array} \]

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 (* x (/ y a)))

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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
    code = x * (y / a)
end function

Java

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

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return x * (y / a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(x * Float64(y / a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = x * (y / a);
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_] := N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in x around inf 53.7%

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

   1. associate-*l/ 51.1%

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

4. Simplified 51.1%

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

   1. associate-*l/ 53.7%

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

   2. clear-num 53.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y}}} \]

6. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y}}} \]
7. Step-by-step derivation

   1. associate-/r* 50.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{x}}{y}}} \]

   2. clear-num 50.8%

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

   3. associate-/r/ 52.9%

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

8. Applied egg-rr 52.9%

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

9. Final simplification 52.9%

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

Alternative 8: 50.7% accurate, 1.8× speedup

Math

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

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 (/ x (/ a y)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return x / (a / y);
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this 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
    code = x / (a / y)
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return x / (a / y);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return x / (a / y)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(x / Float64(a / y))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = x / (a / y);
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_] := N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in x around inf 53.7%

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

   1. associate-*l/ 51.1%

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

4. Simplified 51.1%

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

   1. associate-/r/ 52.4%

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

6. Applied egg-rr 52.4%

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

7. Final simplification 52.4%

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

Developer target: 91.5% accurate, 0.6× 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 2023332 
(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))