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

Percentage Accurate: 91.2% → 96.5%

Time: 8.5s

Alternatives: 8

Speedup: 0.4×

• 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: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+246}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (- (* x (/ y a)) (* z (/ t a)))
     (if (<= t_1 1e+246)
       (/ (fma x y (* z (- t))) a)
       (* z (/ (- (* x (/ y z)) t) a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+246) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = z * (((x * (y / z)) - t) / a);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	elseif (t_1 <= 1e+246)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+246], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.8%

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

      1. div-sub 63.8%

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

      2. associate-/l* 77.1%

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

      3. associate-/l* 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000007e246

   1. Initial program 98.2%

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

      1. div-sub 97.2%

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

      2. *-commutative 97.2%

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

      3. div-sub 98.2%

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

      4. *-commutative 98.2%

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

      5. fma-neg 98.2%

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

      6. distribute-rgt-neg-out 98.2%

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

   3. Simplified 98.2%

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

   if 1.00000000000000007e246 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 76.0%

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

      1. div-sub 69.7%

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

      2. associate-/l* 69.6%

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

      3. associate-/l* 90.6%

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-/r* 87.8%

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

      3. div-sub 91.0%

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

      4. associate-/l* 93.9%

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

   7. Simplified 93.9%

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

Alternative 2: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{-1}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+28)
   (* x (/ y a))
   (if (<= (* x y) -0.005)
     (* (* z t) (/ -1.0 a))
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (/ (* x y) a)
       (* t (/ z (- a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * (y / a);
	} else if ((x * y) <= -0.005) {
		tmp = (z * t) * (-1.0 / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / a;
	} else {
		tmp = t * (z / -a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-5d+28)) then
        tmp = x * (y / a)
    else if ((x * y) <= (-0.005d0)) then
        tmp = (z * t) * ((-1.0d0) / a)
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) / a
    else
        tmp = t * (z / -a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * (y / a);
	} else if ((x * y) <= -0.005) {
		tmp = (z * t) * (-1.0 / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / a;
	} else {
		tmp = t * (z / -a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+28:
		tmp = x * (y / a)
	elif (x * y) <= -0.005:
		tmp = (z * t) * (-1.0 / a)
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) / a
	else:
		tmp = t * (z / -a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+28)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(Float64(z * t) * Float64(-1.0 / a));
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(t * Float64(z / Float64(-a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+28)
		tmp = x * (y / a);
	elseif ((x * y) <= -0.005)
		tmp = (z * t) * (-1.0 / a);
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) / a;
	else
		tmp = t * (z / -a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+28], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(N[(z * t), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(t * N[(z / (-a)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.99999999999999957e28

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf 68.5%

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

      1. associate-*r/ 74.2%

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

   5. Simplified 74.2%

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

   if -4.99999999999999957e28 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. mul-1-neg 76.7%

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

   5. Simplified 76.7%

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

      1. add-sqr-sqrt 61.7%

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

      2. sqrt-unprod 63.1%

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

      3. sqr-neg 63.1%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 1.3%

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

      6. associate-*l/ 0.9%

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

      7. frac-2neg 0.9%

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

   7. Applied egg-rr 0.9%

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

      1. add-sqr-sqrt 0.3%

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

      2. sqrt-unprod 29.9%

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

      3. sqr-neg 29.9%

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

      4. sqrt-unprod 43.0%

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

      5. add-sqr-sqrt 63.4%

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

      6. distribute-rgt-neg-out 63.4%

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

      7. associate-/r/ 63.9%

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

      8. clear-num 64.1%

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

      9. distribute-neg-frac 64.1%

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

      10. metadata-eval 64.1%

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

      11. associate-/l/ 76.7%

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

   9. Applied egg-rr 76.7%

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

       1. associate-/r/ 76.9%

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

   11. Applied egg-rr 76.9%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf 70.3%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. mul-1-neg 84.2%

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

      2. associate-/l* 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

      4. distribute-neg-frac2 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{-1}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+28)
   (* x (/ y a))
   (if (<= (* x y) -0.005)
     (/ (* z t) (- a))
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (/ (* x y) a)
       (* t (/ z (- a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * (y / a);
	} else if ((x * y) <= -0.005) {
		tmp = (z * t) / -a;
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / a;
	} else {
		tmp = t * (z / -a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-5d+28)) then
        tmp = x * (y / a)
    else if ((x * y) <= (-0.005d0)) then
        tmp = (z * t) / -a
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) / a
    else
        tmp = t * (z / -a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * (y / a);
	} else if ((x * y) <= -0.005) {
		tmp = (z * t) / -a;
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / a;
	} else {
		tmp = t * (z / -a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+28:
		tmp = x * (y / a)
	elif (x * y) <= -0.005:
		tmp = (z * t) / -a
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) / a
	else:
		tmp = t * (z / -a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+28)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(Float64(z * t) / Float64(-a));
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(t * Float64(z / Float64(-a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+28)
		tmp = x * (y / a);
	elseif ((x * y) <= -0.005)
		tmp = (z * t) / -a;
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) / a;
	else
		tmp = t * (z / -a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+28], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(N[(z * t), $MachinePrecision] / (-a)), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(t * N[(z / (-a)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.99999999999999957e28

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf 68.5%

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

      1. associate-*r/ 74.2%

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

   5. Simplified 74.2%

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

   if -4.99999999999999957e28 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. mul-1-neg 76.7%

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

   5. Simplified 76.7%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf 70.3%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. mul-1-neg 84.2%

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

      2. associate-/l* 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

      4. distribute-neg-frac2 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+246}\right):\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+246)))
     (* z (/ (- (* x (/ y z)) t) a))
     (/ t_1 a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+246)) {
		tmp = z * (((x * (y / z)) - t) / a);
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+246)) {
		tmp = z * (((x * (y / z)) - t) / a);
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+246):
		tmp = z * (((x * (y / z)) - t) / a)
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+246))
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+246)))
		tmp = z * (((x * (y / z)) - t) / a);
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+246]], $MachinePrecision]], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1.00000000000000007e246 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 71.3%

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

      1. div-sub 67.5%

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

      2. associate-/l* 72.5%

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

      3. associate-/l* 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in z around inf 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-/r* 87.4%

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

      3. div-sub 89.3%

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

      4. associate-/l* 91.1%

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

   7. Simplified 91.1%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000007e246

   1. Initial program 98.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+246}\right):\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+246}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (- (* x (/ y a)) (* z (/ t a)))
     (if (<= t_1 1e+246) (/ t_1 a) (* z (/ (- (* x (/ y z)) t) a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+246) {
		tmp = t_1 / a;
	} else {
		tmp = z * (((x * (y / z)) - t) / a);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+246) {
		tmp = t_1 / a;
	} else {
		tmp = z * (((x * (y / z)) - t) / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x * (y / a)) - (z * (t / a))
	elif t_1 <= 1e+246:
		tmp = t_1 / a
	else:
		tmp = z * (((x * (y / z)) - t) / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	elseif (t_1 <= 1e+246)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x * (y / a)) - (z * (t / a));
	elseif (t_1 <= 1e+246)
		tmp = t_1 / a;
	else
		tmp = z * (((x * (y / z)) - t) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+246], N[(t$95$1 / a), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.8%

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

      1. div-sub 63.8%

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

      2. associate-/l* 77.1%

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

      3. associate-/l* 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000007e246

   1. Initial program 98.2%

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

   if 1.00000000000000007e246 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 76.0%

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

      1. div-sub 69.7%

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

      2. associate-/l* 69.6%

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

      3. associate-/l* 90.6%

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in z around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. associate-/r* 87.8%

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

      3. div-sub 91.0%

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

      4. associate-/l* 93.9%

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

   7. Simplified 93.9%

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

Alternative 6: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* t (/ z (- a)))
   (if (<= (* z t) 5e+284) (/ (- (* x y) (* z t)) a) (/ t (/ a (- z))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t * (z / -a);
	} else if ((z * t) <= 5e+284) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t / (a / -z);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t * (z / -a);
	} else if ((z * t) <= 5e+284) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t / (a / -z);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t * (z / -a)
	elif (z * t) <= 5e+284:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t / (a / -z)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(t * Float64(z / Float64(-a)));
	elseif (Float64(z * t) <= 5e+284)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(t / Float64(a / Float64(-z)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t * (z / -a);
	elseif ((z * t) <= 5e+284)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t / (a / -z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(t * N[(z / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+284], 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 39.2%

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

   3. Taylor expanded in x around 0 39.2%

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

      1. mul-1-neg 39.2%

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

      2. associate-/l* 90.8%

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

      3. distribute-rgt-neg-in 90.8%

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

      4. distribute-neg-frac2 90.8%

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

   5. Simplified 90.8%

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

   if -inf.0 < (*.f64 z t) < 4.9999999999999999e284

   1. Initial program 96.8%

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

   if 4.9999999999999999e284 < (*.f64 z t)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. associate-*r* 67.3%

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

      2. mul-1-neg 67.3%

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

   5. Simplified 67.3%

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

      1. add-sqr-sqrt 43.1%

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

      2. sqrt-unprod 42.9%

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

      3. sqr-neg 42.9%

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

      4. sqrt-unprod 0.1%

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

      5. add-sqr-sqrt 0.1%

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

      6. associate-*l/ 0.2%

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

      7. frac-2neg 0.2%

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

   7. Applied egg-rr 0.2%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 43.6%

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

      3. sqr-neg 43.6%

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

      4. sqrt-unprod 57.0%

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

      5. add-sqr-sqrt 99.8%

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

      6. distribute-rgt-neg-out 99.8%

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

      7. associate-/r/ 99.8%

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

      8. distribute-neg-frac 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-78}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e-109)
   (* x (/ y a))
   (if (<= (* x y) 1e-78) (* t (/ z (- a))) (/ (* x y) a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e-109) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-78) {
		tmp = t * (z / -a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-5d-109)) then
        tmp = x * (y / a)
    else if ((x * y) <= 1d-78) then
        tmp = t * (z / -a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e-109) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-78) {
		tmp = t * (z / -a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e-109:
		tmp = x * (y / a)
	elif (x * y) <= 1e-78:
		tmp = t * (z / -a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e-109)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= 1e-78)
		tmp = Float64(t * Float64(z / Float64(-a)));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e-109)
		tmp = x * (y / a);
	elseif ((x * y) <= 1e-78)
		tmp = t * (z / -a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-78], N[(t * N[(z / (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.0000000000000002e-109

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 62.3%

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

      1. associate-*r/ 62.5%

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

   5. Simplified 62.5%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. mul-1-neg 84.2%

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

      2. associate-/l* 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

      4. distribute-neg-frac2 83.2%

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

   5. Simplified 83.2%

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

   if 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf 71.7%

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

Alternative 8: 51.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x * (y / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

3. Taylor expanded in x around inf 50.4%

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

   1. associate-*r/ 50.2%

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

5. Simplified 50.2%

   \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]
6. Add Preprocessing

Developer target: 91.6% accurate, 0.4× 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 2024091 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (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))