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

Percentage Accurate: 91.3% → 94.2%

Time: 9.3s

Alternatives: 8

Speedup: 0.6×

• 28.0% 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: 91.3% 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: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ t_2 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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)) (t_2 (* z (/ (- t) a))))
   (if (<= (* z t) (- INFINITY))
     t_2
     (if (<= (* z t) -5e-223)
       t_1
       (if (<= (* z t) 4e-252)
         (* x (/ y a))
         (if (<= (* z t) 5e+218) t_1 t_2))))))

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 t_2 = z * (-t / a);
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_2;
	} else if ((z * t) <= -5e-223) {
		tmp = t_1;
	} else if ((z * t) <= 4e-252) {
		tmp = x * (y / a);
	} else if ((z * t) <= 5e+218) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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 t_2 = z * (-t / a);
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if ((z * t) <= -5e-223) {
		tmp = t_1;
	} else if ((z * t) <= 4e-252) {
		tmp = x * (y / a);
	} else if ((z * t) <= 5e+218) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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
	t_2 = z * (-t / a)
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t_2
	elif (z * t) <= -5e-223:
		tmp = t_1
	elif (z * t) <= 4e-252:
		tmp = x * (y / a)
	elif (z * t) <= 5e+218:
		tmp = t_1
	else:
		tmp = t_2
	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)
	t_2 = Float64(z * Float64(Float64(-t) / a))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_2;
	elseif (Float64(z * t) <= -5e-223)
		tmp = t_1;
	elseif (Float64(z * t) <= 4e-252)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(z * t) <= 5e+218)
		tmp = t_1;
	else
		tmp = t_2;
	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;
	t_2 = z * (-t / a);
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t_2;
	elseif ((z * t) <= -5e-223)
		tmp = t_1;
	elseif ((z * t) <= 4e-252)
		tmp = x * (y / a);
	elseif ((z * t) <= 5e+218)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -5e-223], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e-252], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+218], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0 or 4.99999999999999983e218 < (*.f64 z t)

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-*l/ 92.9%

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

      3. associate-*l* 92.9%

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

      4. *-commutative 92.9%

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

      5. neg-mul-1 92.9%

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

   4. Simplified 92.9%

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

   if -inf.0 < (*.f64 z t) < -5.00000000000000024e-223 or 3.99999999999999977e-252 < (*.f64 z t) < 4.99999999999999983e218

   1. Initial program 96.9%

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

   if -5.00000000000000024e-223 < (*.f64 z t) < 3.99999999999999977e-252

   1. Initial program 83.6%

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

      1. frac-2neg 83.6%

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

      2. neg-sub0 83.6%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

      3. sub-neg 83.6%

         \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

      4. +-commutative 83.6%

         \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

      5. associate--r+ 83.6%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

      6. neg-sub0 83.6%

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

      7. remove-double-neg 83.6%

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

      8. div-sub 83.6%

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

   3. Applied egg-rr 83.6%

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

      1. associate-/l* 86.4%

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

      2. associate-/l* 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in z around 0 81.9%

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

      1. associate-*r/ 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-223}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 2: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+51} \lor \neg \left(a \leq 1.05 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

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 (or (<= a -7.8e+51) (not (<= a 1.05e+70)))
   (- (* x (/ y a)) (* z (/ t a)))
   (/ (- (* x y) (* z t)) a)))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.8e+51) || !(a <= 1.05e+70)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else {
		tmp = ((x * y) - (z * t)) / 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 ((a <= (-7.8d+51)) .or. (.not. (a <= 1.05d+70))) then
        tmp = (x * (y / a)) - (z * (t / a))
    else
        tmp = ((x * y) - (z * t)) / 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 ((a <= -7.8e+51) || !(a <= 1.05e+70)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7.8e+51) or not (a <= 1.05e+70):
		tmp = (x * (y / a)) - (z * (t / a))
	else:
		tmp = ((x * y) - (z * t)) / 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 ((a <= -7.8e+51) || !(a <= 1.05e+70))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / 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 ((a <= -7.8e+51) || ~((a <= 1.05e+70)))
		tmp = (x * (y / a)) - (z * (t / a));
	else
		tmp = ((x * y) - (z * t)) / 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[Or[LessEqual[a, -7.8e+51], N[Not[LessEqual[a, 1.05e+70]], $MachinePrecision]], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999968e51 or 1.05000000000000004e70 < a

   1. Initial program 76.4%

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

      1. div-sub 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-/l* 85.0%

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

      4. associate-/l* 92.6%

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

      5. frac-sub 35.7%

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

      6. div-sub 35.7%

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

   3. Applied egg-rr 35.7%

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

      1. times-frac 43.4%

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

      2. *-inverses 55.3%

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

      3. associate-*l/ 55.3%

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

      4. *-rgt-identity 55.3%

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

      5. associate-/r/ 54.5%

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

      6. associate-/l* 59.9%

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

      7. associate-*r/ 47.6%

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

      8. associate-/l/ 44.0%

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

      9. associate-/l* 41.9%

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

      10. times-frac 61.9%

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

      11. *-inverses 85.0%

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

      12. associate-*r/ 85.0%

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

      13. *-lft-identity 85.0%

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

      14. associate-*r/ 92.7%

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

   5. Simplified 92.7%

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

   if -7.79999999999999968e51 < a < 1.05000000000000004e70

   1. Initial program 95.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+51} \lor \neg \left(a \leq 1.05 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 3: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}} - \frac{x}{\frac{-a}{y}}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \end{array} \end{array} \]

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 (<= a -1.25e+52)
   (- (/ z (/ (- a) t)) (/ x (/ (- a) y)))
   (if (<= a 3.8e+69)
     (/ (- (* x y) (* z t)) a)
     (- (* x (/ y a)) (* z (/ t a))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.25e+52) {
		tmp = (z / (-a / t)) - (x / (-a / y));
	} else if (a <= 3.8e+69) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (x * (y / a)) - (z * (t / 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 (a <= (-1.25d+52)) then
        tmp = (z / (-a / t)) - (x / (-a / y))
    else if (a <= 3.8d+69) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = (x * (y / a)) - (z * (t / 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 (a <= -1.25e+52) {
		tmp = (z / (-a / t)) - (x / (-a / y));
	} else if (a <= 3.8e+69) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = (x * (y / a)) - (z * (t / a));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.25e+52:
		tmp = (z / (-a / t)) - (x / (-a / y))
	elif a <= 3.8e+69:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = (x * (y / a)) - (z * (t / 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 (a <= -1.25e+52)
		tmp = Float64(Float64(z / Float64(Float64(-a) / t)) - Float64(x / Float64(Float64(-a) / y)));
	elseif (a <= 3.8e+69)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / 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 (a <= -1.25e+52)
		tmp = (z / (-a / t)) - (x / (-a / y));
	elseif (a <= 3.8e+69)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = (x * (y / a)) - (z * (t / 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[a, -1.25e+52], N[(N[(z / N[((-a) / t), $MachinePrecision]), $MachinePrecision] - N[(x / N[((-a) / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.8e+69], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.25e52

   1. Initial program 76.7%

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

      1. frac-2neg 76.7%

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

      2. neg-sub0 76.7%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

      3. sub-neg 76.7%

         \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

      4. +-commutative 76.7%

         \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

      5. associate--r+ 76.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

      6. neg-sub0 76.7%

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

      7. remove-double-neg 76.7%

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

      8. div-sub 76.7%

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

   3. Applied egg-rr 76.7%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 96.4%

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

   5. Simplified 96.4%

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

   if -1.25e52 < a < 3.80000000000000028e69

   1. Initial program 95.6%

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

   if 3.80000000000000028e69 < a

   1. Initial program 76.0%

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

      1. div-sub 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/l* 86.8%

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

      4. associate-/l* 90.4%

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

      5. frac-sub 20.5%

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

      6. div-sub 20.5%

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

   3. Applied egg-rr 20.5%

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

      1. times-frac 31.2%

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

      2. *-inverses 48.5%

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

      3. associate-*l/ 48.5%

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

      4. *-rgt-identity 48.5%

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

      5. associate-/r/ 46.8%

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

      6. associate-/l* 56.2%

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

      7. associate-*r/ 48.7%

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

      8. associate-/l/ 44.9%

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

      9. associate-/l* 42.9%

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

      10. times-frac 61.9%

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

      11. *-inverses 85.0%

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

      12. associate-*r/ 85.0%

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

      13. *-lft-identity 85.0%

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

      14. associate-*r/ 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}} - \frac{x}{\frac{-a}{y}}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \end{array} \]

Alternative 4: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 (<= (* x y) -5e+67)
   (/ x (/ a y))
   (if (<= (* x y) 5e-13) (* z (/ (- t) 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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = z * (-t / 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 ((x * y) <= (-5d+67)) then
        tmp = x / (a / y)
    else if ((x * y) <= 5d-13) then
        tmp = z * (-t / 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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = z * (-t / 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 (x * y) <= -5e+67:
		tmp = x / (a / y)
	elif (x * y) <= 5e-13:
		tmp = z * (-t / 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 (Float64(x * y) <= -5e+67)
		tmp = Float64(x / Float64(a / y));
	elseif (Float64(x * y) <= 5e-13)
		tmp = Float64(z * Float64(Float64(-t) / 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 ((x * y) <= -5e+67)
		tmp = x / (a / y);
	elseif ((x * y) <= 5e-13)
		tmp = z * (-t / 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[N[(x * y), $MachinePrecision], -5e+67], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-13], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999976e67

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf 78.5%

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

      1. associate-*l/ 84.5%

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

      2. *-commutative 84.5%

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

   4. Simplified 84.5%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-/l* 82.7%

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

   6. Applied egg-rr 82.7%

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

   if -4.99999999999999976e67 < (*.f64 x y) < 4.9999999999999999e-13

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 72.7%

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

      3. associate-*l* 72.7%

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

      4. *-commutative 72.7%

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

      5. neg-mul-1 72.7%

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

   4. Simplified 72.7%

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

   if 4.9999999999999999e-13 < (*.f64 x y)

   1. Initial program 85.9%

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

      1. frac-2neg 85.9%

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

      2. neg-sub0 85.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

      3. sub-neg 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

      4. +-commutative 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

      5. associate--r+ 85.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

      6. neg-sub0 85.9%

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

      7. remove-double-neg 85.9%

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

      8. div-sub 84.6%

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

   3. Applied egg-rr 84.6%

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

      1. associate-/l* 79.4%

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

      2. associate-/l* 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in z around 0 65.6%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 (<= (* x y) -5e+67)
   (/ x (/ a y))
   (if (<= (* x y) 5e-13) (* t (/ (- z) 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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = t * (-z / 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 ((x * y) <= (-5d+67)) then
        tmp = x / (a / y)
    else if ((x * y) <= 5d-13) then
        tmp = t * (-z / 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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = t * (-z / 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 (x * y) <= -5e+67:
		tmp = x / (a / y)
	elif (x * y) <= 5e-13:
		tmp = t * (-z / 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 (Float64(x * y) <= -5e+67)
		tmp = Float64(x / Float64(a / y));
	elseif (Float64(x * y) <= 5e-13)
		tmp = Float64(t * Float64(Float64(-z) / 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 ((x * y) <= -5e+67)
		tmp = x / (a / y);
	elseif ((x * y) <= 5e-13)
		tmp = t * (-z / 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[N[(x * y), $MachinePrecision], -5e+67], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-13], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999976e67

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf 78.5%

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

      1. associate-*l/ 84.5%

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

      2. *-commutative 84.5%

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

   4. Simplified 84.5%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-/l* 82.7%

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

   6. Applied egg-rr 82.7%

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

   if -4.99999999999999976e67 < (*.f64 x y) < 4.9999999999999999e-13

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 72.7%

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

      3. associate-*l* 72.7%

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

      4. *-commutative 72.7%

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

      5. neg-mul-1 72.7%

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

   4. Simplified 72.7%

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

      1. associate-*l/ 71.7%

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

      2. associate-/l* 78.1%

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

      3. remove-double-neg 78.1%

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

      4. frac-2neg 78.1%

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

      5. clear-num 78.0%

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

      6. associate-/r* 71.6%

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

      7. neg-mul-1 71.6%

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

      8. metadata-eval 71.6%

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

      9. *-commutative 71.6%

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

      10. times-frac 78.0%

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

      11. associate-/r* 78.0%

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

      12. clear-num 78.1%

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

      13. div-inv 78.1%

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

      14. metadata-eval 78.1%

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

      15. metadata-eval 78.1%

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

      16. metadata-eval 78.1%

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

      17. *-commutative 78.1%

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

      18. metadata-eval 78.1%

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

      19. neg-mul-1 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. clear-num 78.0%

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

      2. associate-/r/ 78.5%

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

      3. *-commutative 78.5%

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

      4. /-rgt-identity 78.5%

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

   8. Applied egg-rr 78.5%

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

   if 4.9999999999999999e-13 < (*.f64 x y)

   1. Initial program 85.9%

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

      1. frac-2neg 85.9%

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

      2. neg-sub0 85.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

      3. sub-neg 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

      4. +-commutative 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

      5. associate--r+ 85.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

      6. neg-sub0 85.9%

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

      7. remove-double-neg 85.9%

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

      8. div-sub 84.6%

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

   3. Applied egg-rr 84.6%

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

      1. associate-/l* 79.4%

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

      2. associate-/l* 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in z around 0 65.6%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 (<= (* x y) -5e+67)
   (/ x (/ a y))
   (if (<= (* x y) 5e-13) (/ (- t) (/ a z)) (* 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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = -t / (a / z);
	} 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 ((x * y) <= (-5d+67)) then
        tmp = x / (a / y)
    else if ((x * y) <= 5d-13) then
        tmp = -t / (a / z)
    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 ((x * y) <= -5e+67) {
		tmp = x / (a / y);
	} else if ((x * y) <= 5e-13) {
		tmp = -t / (a / z);
	} 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 (x * y) <= -5e+67:
		tmp = x / (a / y)
	elif (x * y) <= 5e-13:
		tmp = -t / (a / z)
	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 (Float64(x * y) <= -5e+67)
		tmp = Float64(x / Float64(a / y));
	elseif (Float64(x * y) <= 5e-13)
		tmp = Float64(Float64(-t) / Float64(a / z));
	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 ((x * y) <= -5e+67)
		tmp = x / (a / y);
	elseif ((x * y) <= 5e-13)
		tmp = -t / (a / z);
	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[N[(x * y), $MachinePrecision], -5e+67], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-13], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999976e67

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf 78.5%

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

      1. associate-*l/ 84.5%

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

      2. *-commutative 84.5%

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

   4. Simplified 84.5%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-/l* 82.7%

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

   6. Applied egg-rr 82.7%

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

   if -4.99999999999999976e67 < (*.f64 x y) < 4.9999999999999999e-13

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 72.7%

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

      3. associate-*l* 72.7%

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

      4. *-commutative 72.7%

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

      5. neg-mul-1 72.7%

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

   4. Simplified 72.7%

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

      1. associate-*l/ 71.7%

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

      2. associate-/l* 78.1%

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

      3. remove-double-neg 78.1%

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

      4. frac-2neg 78.1%

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

      5. clear-num 78.0%

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

      6. associate-/r* 71.6%

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

      7. neg-mul-1 71.6%

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

      8. metadata-eval 71.6%

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

      9. *-commutative 71.6%

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

      10. times-frac 78.0%

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

      11. associate-/r* 78.0%

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

      12. clear-num 78.1%

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

      13. div-inv 78.1%

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

      14. metadata-eval 78.1%

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

      15. metadata-eval 78.1%

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

      16. metadata-eval 78.1%

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

      17. *-commutative 78.1%

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

      18. metadata-eval 78.1%

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

      19. neg-mul-1 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 4.9999999999999999e-13 < (*.f64 x y)

   1. Initial program 85.9%

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

      1. frac-2neg 85.9%

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

      2. neg-sub0 85.9%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

      3. sub-neg 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

      4. +-commutative 85.9%

         \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

      5. associate--r+ 85.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

      6. neg-sub0 85.9%

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

      7. remove-double-neg 85.9%

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

      8. div-sub 84.6%

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

   3. Applied egg-rr 84.6%

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

      1. associate-/l* 79.4%

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

      2. associate-/l* 84.6%

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

   5. Simplified 84.6%

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

   6. Taylor expanded in z around 0 65.6%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7: 51.5% accurate, 1.8× speedup

Math

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

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 (* x (/ y a)))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 87.4%

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

   1. frac-2neg 87.4%

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

   2. neg-sub0 87.4%

      \[\leadsto \frac{\color{blue}{0 - \left(x \cdot y - z \cdot t\right)}}{-a} \]

   3. sub-neg 87.4%

      \[\leadsto \frac{0 - \color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)}}{-a} \]

   4. +-commutative 87.4%

      \[\leadsto \frac{0 - \color{blue}{\left(\left(-z \cdot t\right) + x \cdot y\right)}}{-a} \]

   5. associate--r+ 87.4%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(-z \cdot t\right)\right) - x \cdot y}}{-a} \]

   6. neg-sub0 87.4%

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

   7. remove-double-neg 87.4%

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

   8. div-sub 85.9%

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

3. Applied egg-rr 85.9%

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

   1. associate-/l* 85.0%

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

   2. associate-/l* 85.8%

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

5. Simplified 85.8%

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

6. Taylor expanded in z around 0 48.2%

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

   1. associate-*r/ 50.5%

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

8. Simplified 50.5%

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

9. Final simplification 50.5%

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

Alternative 8: 51.3% accurate, 1.8× speedup

Math

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

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 (/ x (/ a y)))

C

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

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 = x / (a / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 87.4%

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

2. Taylor expanded in x around inf 48.2%

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

   1. associate-*l/ 52.9%

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

   2. *-commutative 52.9%

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

4. Simplified 52.9%

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

   1. associate-*r/ 48.2%

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

   2. *-commutative 48.2%

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

   3. associate-/l* 51.0%

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

6. Applied egg-rr 51.0%

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

7. Final simplification 51.0%

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

Developer target: 92.0% 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 2023297 
(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))