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

Percentage Accurate: 91.1% → 96.9%

Time: 7.7s

Alternatives: 11

Speedup: 0.5×

• 27.7% 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: 91.1% 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: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+274} \lor \neg \left(t_1 \leq 10^{+288}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 -5e+274) (not (<= t_1 1e+288)))
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (/ t_1 a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -5e+274) || !(t_1 <= 1e+288)) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -5e+274) || !(t_1 <= 1e+288))
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+274], N[Not[LessEqual[t$95$1, 1e+288]], $MachinePrecision]], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.9999999999999998e274 or 1e288 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 73.3%

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

   2. Taylor expanded in x around 0 71.8%

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

      1. fma-def 71.8%

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

      2. associate-/l* 82.3%

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

      3. associate-/l* 94.3%

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

   4. Simplified 94.3%

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

   if -4.9999999999999998e274 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e288

   1. Initial program 98.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+274} \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+288}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 2: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-44)
   (/ x (/ a y))
   (if (<= (* x y) 2e-133)
     (/ (- t) (/ a z))
     (if (<= (* x y) 5e+59)
       (/ (* x y) a)
       (if (<= (* x y) 1e+130) (- (* (/ t a) z)) (* y (/ x a)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x / (a / y);
	} else if ((x * y) <= 2e-133) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 1e+130) {
		tmp = -((t / a) * z);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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-44)) then
        tmp = x / (a / y)
    else if ((x * y) <= 2d-133) then
        tmp = -t / (a / z)
    else if ((x * y) <= 5d+59) then
        tmp = (x * y) / a
    else if ((x * y) <= 1d+130) then
        tmp = -((t / a) * z)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x / (a / y);
	} else if ((x * y) <= 2e-133) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 1e+130) {
		tmp = -((t / a) * z);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-44:
		tmp = x / (a / y)
	elif (x * y) <= 2e-133:
		tmp = -t / (a / z)
	elif (x * y) <= 5e+59:
		tmp = (x * y) / a
	elif (x * y) <= 1e+130:
		tmp = -((t / a) * z)
	else:
		tmp = y * (x / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-44)
		tmp = Float64(x / Float64(a / y));
	elseif (Float64(x * y) <= 2e-133)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (Float64(x * y) <= 5e+59)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 1e+130)
		tmp = Float64(-Float64(Float64(t / a) * z));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-44)
		tmp = x / (a / y);
	elseif ((x * y) <= 2e-133)
		tmp = -t / (a / z);
	elseif ((x * y) <= 5e+59)
		tmp = (x * y) / a;
	elseif ((x * y) <= 1e+130)
		tmp = -((t / a) * z);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-44], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-133], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+59], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999991e-44

   1. Initial program 90.4%

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

   2. Taylor expanded in x around inf 69.9%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

      1. associate-*r/ 69.9%

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

      2. *-commutative 69.9%

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

      3. associate-/l* 71.3%

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

   6. Applied egg-rr 71.3%

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

   if -1.99999999999999991e-44 < (*.f64 x y) < 2.0000000000000001e-133

   1. Initial program 90.5%

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

      1. div-inv 90.4%

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

      2. fma-neg 90.4%

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

      3. *-commutative 90.4%

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

      4. distribute-rgt-neg-in 90.4%

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

   3. Applied egg-rr 90.4%

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

   4. Taylor expanded in x around 0 90.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-def 90.4%

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

      4. fma-neg 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in y around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. associate-/l* 84.2%

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

      3. distribute-neg-frac 84.2%

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

   9. Simplified 84.2%

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

   if 2.0000000000000001e-133 < (*.f64 x y) < 4.9999999999999997e59

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 65.5%

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

   if 4.9999999999999997e59 < (*.f64 x y) < 1.0000000000000001e130

   1. Initial program 89.1%

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

      1. div-inv 89.3%

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

      2. fma-neg 89.3%

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

      3. *-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

   3. Applied egg-rr 89.3%

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

   4. Taylor expanded in x around 0 89.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 89.3%

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

      2. +-commutative 89.3%

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

      3. fma-def 89.3%

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

      4. fma-neg 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in y around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. associate-*r* 63.6%

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

      3. neg-mul-1 63.6%

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

      4. *-commutative 63.6%

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

      5. *-rgt-identity 63.6%

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

      6. associate-*r/ 63.6%

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

      7. associate-*l* 74.0%

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

      8. associate-*r/ 74.2%

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

      9. *-rgt-identity 74.2%

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

   9. Simplified 74.2%

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

   if 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 92.0%

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

   2. Taylor expanded in x around inf 82.7%

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

      1. associate-*r/ 84.5%

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

   4. Simplified 84.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* z t)) a)))
   (if (<= t_1 5e+261) t_1 (- (* y (/ x a)) (/ t (/ a z))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= 5e+261) {
		tmp = t_1;
	} else {
		tmp = (y * (x / a)) - (t / (a / z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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) :: t_1
    real(8) :: tmp
    t_1 = ((x * y) - (z * t)) / a
    if (t_1 <= 5d+261) then
        tmp = t_1
    else
        tmp = (y * (x / a)) - (t / (a / z))
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= 5e+261) {
		tmp = t_1;
	} else {
		tmp = (y * (x / a)) - (t / (a / z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = ((x * y) - (z * t)) / a
	tmp = 0
	if t_1 <= 5e+261:
		tmp = t_1
	else:
		tmp = (y * (x / a)) - (t / (a / z))
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= 5e+261)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t / Float64(a / z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x * y) - (z * t)) / a;
	tmp = 0.0;
	if (t_1 <= 5e+261)
		tmp = t_1;
	else
		tmp = (y * (x / a)) - (t / (a / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+261], t$95$1, N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 5.0000000000000001e261

   1. Initial program 95.1%

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

   if 5.0000000000000001e261 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 76.7%

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

      1. div-inv 76.6%

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

      2. fma-neg 80.7%

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

      3. *-commutative 80.7%

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

      4. distribute-rgt-neg-in 80.7%

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

   3. Applied egg-rr 80.7%

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

   4. Taylor expanded in x around 0 76.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.6%

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

      2. +-commutative 76.6%

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

      3. fma-def 80.7%

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

      4. fma-neg 76.6%

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

   6. Simplified 76.6%

      \[\leadsto \color{blue}{\left(y \cdot x - t \cdot z\right)} \cdot \frac{1}{a} \]
   7. Step-by-step derivation

      1. un-div-inv 76.7%

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

      2. div-sub 76.7%

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

      3. associate-*l/ 84.5%

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

      4. associate-/r/ 84.4%

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

      5. div-inv 84.4%

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

      6. clear-num 84.5%

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

      7. associate-/l* 89.8%

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

   8. Applied egg-rr 89.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 4: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (- (* (/ t a) z))
   (if (<= (* z t) 5e+279) (/ (- (* x y) (* z t)) a) (/ (- t) (/ a z)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = -((t / a) * z);
	} else if ((z * t) <= 5e+279) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = -((t / a) * z);
	} else if ((z * t) <= 5e+279) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = -((t / a) * z)
	elif (z * t) <= 5e+279:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -t / (a / z)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(-Float64(Float64(t / a) * z));
	elseif (Float64(z * t) <= 5e+279)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = -((t / a) * z);
	elseif ((z * t) <= 5e+279)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -t / (a / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), If[LessEqual[N[(z * t), $MachinePrecision], 5e+279], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 63.3%

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

      1. div-inv 63.3%

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

      2. fma-neg 68.9%

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

      3. *-commutative 68.9%

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

      4. distribute-rgt-neg-in 68.9%

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

   3. Applied egg-rr 68.9%

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

   4. Taylor expanded in x around 0 63.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 63.3%

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

      2. +-commutative 63.3%

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

      3. fma-def 68.9%

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

      4. fma-neg 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in y around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. associate-*r* 68.9%

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

      3. neg-mul-1 68.9%

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

      4. *-commutative 68.9%

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

      5. *-rgt-identity 68.9%

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

      6. associate-*r/ 68.9%

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

      7. associate-*l* 94.7%

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

      8. associate-*r/ 94.7%

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

      9. *-rgt-identity 94.7%

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

   9. Simplified 94.7%

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

   if -inf.0 < (*.f64 z t) < 5.0000000000000002e279

   1. Initial program 96.3%

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

   if 5.0000000000000002e279 < (*.f64 z t)

   1. Initial program 52.2%

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

      1. div-inv 52.3%

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

      2. fma-neg 59.4%

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

      3. *-commutative 59.4%

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

      4. distribute-rgt-neg-in 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in x around 0 52.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.3%

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

      2. +-commutative 52.3%

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

      3. fma-def 59.4%

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

      4. fma-neg 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in y around 0 59.3%

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

      1. mul-1-neg 59.3%

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

      2. associate-/l* 92.9%

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

      3. distribute-neg-frac 92.9%

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

   9. Simplified 92.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 5: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) -2e+146)
   (- (/ x (/ a y)) (/ z (/ a t)))
   (if (<= (* z t) 5e+279) (/ (- (* x y) (* z t)) a) (/ (- t) (/ a z)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -2e+146) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if ((z * t) <= 5e+279) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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) <= (-2d+146)) then
        tmp = (x / (a / y)) - (z / (a / t))
    else if ((z * t) <= 5d+279) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = -t / (a / z)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -2e+146) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if ((z * t) <= 5e+279) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -2e+146:
		tmp = (x / (a / y)) - (z / (a / t))
	elif (z * t) <= 5e+279:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -t / (a / z)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= -2e+146)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (Float64(z * t) <= 5e+279)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -2e+146)
		tmp = (x / (a / y)) - (z / (a / t));
	elseif ((z * t) <= 5e+279)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -t / (a / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+146], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+279], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.99999999999999987e146

   1. Initial program 78.4%

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

      1. div-sub 78.3%

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

      2. associate-/l* 83.8%

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

      3. associate-/l* 99.8%

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

   3. Applied egg-rr 99.8%

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

   if -1.99999999999999987e146 < (*.f64 z t) < 5.0000000000000002e279

   1. Initial program 96.5%

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

   if 5.0000000000000002e279 < (*.f64 z t)

   1. Initial program 52.2%

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

      1. div-inv 52.3%

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

      2. fma-neg 59.4%

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

      3. *-commutative 59.4%

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

      4. distribute-rgt-neg-in 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in x around 0 52.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.3%

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

      2. +-commutative 52.3%

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

      3. fma-def 59.4%

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

      4. fma-neg 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in y around 0 59.3%

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

      1. mul-1-neg 59.3%

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

      2. associate-/l* 92.9%

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

      3. distribute-neg-frac 92.9%

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

   9. Simplified 92.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 6: 67.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -28:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (- (/ z a)))))
   (if (<= z -1.4e+103)
     t_1
     (if (<= z -6.5e+70)
       (* y (/ x a))
       (if (<= z -28.0)
         (- (* (/ t a) z))
         (if (<= z 1.75e-131) (/ (* x y) a) t_1))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * -(z / a);
	double tmp;
	if (z <= -1.4e+103) {
		tmp = t_1;
	} else if (z <= -6.5e+70) {
		tmp = y * (x / a);
	} else if (z <= -28.0) {
		tmp = -((t / a) * z);
	} else if (z <= 1.75e-131) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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) :: t_1
    real(8) :: tmp
    t_1 = t * -(z / a)
    if (z <= (-1.4d+103)) then
        tmp = t_1
    else if (z <= (-6.5d+70)) then
        tmp = y * (x / a)
    else if (z <= (-28.0d0)) then
        tmp = -((t / a) * z)
    else if (z <= 1.75d-131) then
        tmp = (x * y) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * -(z / a);
	double tmp;
	if (z <= -1.4e+103) {
		tmp = t_1;
	} else if (z <= -6.5e+70) {
		tmp = y * (x / a);
	} else if (z <= -28.0) {
		tmp = -((t / a) * z);
	} else if (z <= 1.75e-131) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = t * -(z / a)
	tmp = 0
	if z <= -1.4e+103:
		tmp = t_1
	elif z <= -6.5e+70:
		tmp = y * (x / a)
	elif z <= -28.0:
		tmp = -((t / a) * z)
	elif z <= 1.75e-131:
		tmp = (x * y) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(-Float64(z / a)))
	tmp = 0.0
	if (z <= -1.4e+103)
		tmp = t_1;
	elseif (z <= -6.5e+70)
		tmp = Float64(y * Float64(x / a));
	elseif (z <= -28.0)
		tmp = Float64(-Float64(Float64(t / a) * z));
	elseif (z <= 1.75e-131)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * -(z / a);
	tmp = 0.0;
	if (z <= -1.4e+103)
		tmp = t_1;
	elseif (z <= -6.5e+70)
		tmp = y * (x / a);
	elseif (z <= -28.0)
		tmp = -((t / a) * z);
	elseif (z <= 1.75e-131)
		tmp = (x * y) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[z, -1.4e+103], t$95$1, If[LessEqual[z, -6.5e+70], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -28.0], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), If[LessEqual[z, 1.75e-131], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.40000000000000004e103 or 1.7500000000000001e-131 < z

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0 51.0%

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

      1. associate-*r/ 51.0%

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

      2. mul-1-neg 51.0%

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

      3. distribute-rgt-neg-out 51.0%

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

      4. *-commutative 51.0%

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

      5. associate-/l* 53.5%

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

      6. associate-/r/ 55.3%

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

   4. Simplified 55.3%

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

   if -1.40000000000000004e103 < z < -6.49999999999999978e70

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 67.3%

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

      1. associate-*r/ 67.3%

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

   4. Simplified 67.3%

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

   if -6.49999999999999978e70 < z < -28

   1. Initial program 91.7%

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

      1. div-inv 91.6%

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

      2. fma-neg 91.6%

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

      3. *-commutative 91.6%

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

      4. distribute-rgt-neg-in 91.6%

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

   3. Applied egg-rr 91.6%

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

   4. Taylor expanded in x around 0 91.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 91.6%

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

      2. +-commutative 91.6%

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

      3. fma-def 91.6%

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

      4. fma-neg 91.6%

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

   6. Simplified 91.6%

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

   7. Taylor expanded in y around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. associate-*r* 59.5%

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

      3. neg-mul-1 59.5%

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

      4. *-commutative 59.5%

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

      5. *-rgt-identity 59.5%

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

      6. associate-*r/ 59.3%

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

      7. associate-*l* 67.2%

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

      8. associate-*r/ 67.3%

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

      9. *-rgt-identity 67.3%

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

   9. Simplified 67.3%

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

   if -28 < z < 1.7500000000000001e-131

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf 74.1%

      \[\leadsto \color{blue}{\frac{y \cdot x}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+103}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -28:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \end{array} \]

Alternative 7: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+16}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -4.2e-188)
   (* y (/ x a))
   (if (<= y 6.8e+16) (- (* (/ t a) z)) (/ x (/ a y)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -4.2e-188) {
		tmp = y * (x / a);
	} else if (y <= 6.8e+16) {
		tmp = -((t / a) * z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 (y <= (-4.2d-188)) then
        tmp = y * (x / a)
    else if (y <= 6.8d+16) then
        tmp = -((t / a) * z)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -4.2e-188) {
		tmp = y * (x / a);
	} else if (y <= 6.8e+16) {
		tmp = -((t / a) * z);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= -4.2e-188:
		tmp = y * (x / a)
	elif y <= 6.8e+16:
		tmp = -((t / a) * z)
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -4.2e-188)
		tmp = Float64(y * Float64(x / a));
	elseif (y <= 6.8e+16)
		tmp = Float64(-Float64(Float64(t / a) * z));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -4.2e-188)
		tmp = y * (x / a);
	elseif (y <= 6.8e+16)
		tmp = -((t / a) * z);
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -4.2e-188], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+16], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1999999999999998e-188

   1. Initial program 92.2%

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

   2. Taylor expanded in x around inf 59.5%

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

      1. associate-*r/ 59.6%

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

   4. Simplified 59.6%

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

   if -4.1999999999999998e-188 < y < 6.8e16

   1. Initial program 93.0%

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

      1. div-inv 92.8%

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

      2. fma-neg 92.8%

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

      3. *-commutative 92.8%

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

      4. distribute-rgt-neg-in 92.8%

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

   3. Applied egg-rr 92.8%

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

   4. Taylor expanded in x around 0 92.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot z\right) + y \cdot x\right)} \cdot \frac{1}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 92.8%

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

      2. +-commutative 92.8%

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

      3. fma-def 92.8%

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

      4. fma-neg 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in y around 0 63.8%

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

      1. associate-*r/ 63.8%

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

      2. associate-*r* 63.8%

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

      3. neg-mul-1 63.8%

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

      4. *-commutative 63.8%

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

      5. *-rgt-identity 63.8%

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

      6. associate-*r/ 63.7%

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

      7. associate-*l* 66.0%

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

      8. associate-*r/ 65.9%

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

      9. *-rgt-identity 65.9%

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

   9. Simplified 65.9%

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

   if 6.8e16 < y

   1. Initial program 89.1%

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

   2. Taylor expanded in x around inf 67.4%

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

      1. associate-*r/ 69.0%

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

   4. Simplified 69.0%

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

      1. associate-*r/ 67.4%

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

      2. *-commutative 67.4%

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

      3. associate-/l* 68.4%

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

   6. Applied egg-rr 68.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+16}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 8: 51.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-145) (* y (/ x a)) (* x (/ y a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-145) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 <= (-2.2d-145)) then
        tmp = y * (x / a)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-145) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-145:
		tmp = y * (x / a)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-145)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.2e-145)
		tmp = y * (x / a);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-145], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.19999999999999999e-145

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 46.1%

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

   4. Simplified 46.1%

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

   if -2.19999999999999999e-145 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf 63.3%

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

      1. associate-*l/ 61.8%

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

   4. Applied egg-rr 61.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e-142) (/ y (/ a x)) (* x (/ y a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-142) {
		tmp = y / (a / x);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 <= (-2d-142)) then
        tmp = y / (a / x)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e-142) {
		tmp = y / (a / x);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -2e-142:
		tmp = y / (a / x)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e-142)
		tmp = Float64(y / Float64(a / x));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2e-142)
		tmp = y / (a / x);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e-142], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e-142

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf 45.9%

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

      1. associate-/l* 46.8%

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

   4. Simplified 46.8%

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

   if -2.0000000000000001e-142 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf 63.3%

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

      1. associate-*l/ 61.8%

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

   4. Applied egg-rr 61.8%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10: 51.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e+162) (* x (/ y a)) (/ (* x y) a)))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+162) {
		tmp = x * (y / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 <= (-3.3d+162)) then
        tmp = x * (y / a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+162) {
		tmp = x * (y / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e+162:
		tmp = x * (y / a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.3e+162)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.3e+162)
		tmp = x * (y / a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.3e+162], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.29999999999999987e162

   1. Initial program 82.4%

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

   2. Taylor expanded in x around inf 21.6%

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

      1. associate-*l/ 32.2%

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

   4. Applied egg-rr 32.2%

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

   if -3.29999999999999987e162 < z

   1. Initial program 92.7%

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

   2. Taylor expanded in x around inf 61.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 11: 51.3% accurate, 1.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return y * (x / a);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	return y * (x / a);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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 91.6%

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

2. Taylor expanded in x around inf 57.0%

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

   1. associate-*r/ 57.5%

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

4. Simplified 57.5%

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

5. Final simplification 57.5%

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

Developer target: 91.6% 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 2023240 
(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))