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

Percentage Accurate: 91.6% → 94.4%

Time: 6.5s

Alternatives: 10

Speedup: 0.7×

• 28.6% 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: 91.6% 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: 94.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

assert(x < y);
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 = fma(x, (y / a), (-z / (a / t)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
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 = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.6%

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

      1. div-sub 67.8%

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

      2. *-un-lft-identity 67.8%

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

      3. times-frac 75.6%

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

      4. fma-neg 75.6%

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

      5. associate-/l* 91.2%

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

   3. Applied egg-rr 91.2%

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

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

   1. Initial program 97.6%

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

4. Final simplification 96.7%

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

Alternative 2: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \cdot y \leq 6.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 500000000000:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+32}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e+35)
   (/ y (/ a x))
   (if (<= (* x y) 5e-81)
     (/ (* z (- t)) a)
     (if (<= (* x y) 6.6e-23)
       (/ (* x y) a)
       (if (<= (* x y) 500000000000.0)
         (/ (- t) (/ a z))
         (if (<= (* x y) 1e+32)
           (* (* x y) (/ 1.0 a))
           (if (<= (* x y) 5e+93) (/ (- z) (/ a t)) (* x (/ y a)))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+35) {
		tmp = y / (a / x);
	} else if ((x * y) <= 5e-81) {
		tmp = (z * -t) / a;
	} else if ((x * y) <= 6.6e-23) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 500000000000.0) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 1e+32) {
		tmp = (x * y) * (1.0 / a);
	} else if ((x * y) <= 5e+93) {
		tmp = -z / (a / t);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x and y 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) <= (-1d+35)) then
        tmp = y / (a / x)
    else if ((x * y) <= 5d-81) then
        tmp = (z * -t) / a
    else if ((x * y) <= 6.6d-23) then
        tmp = (x * y) / a
    else if ((x * y) <= 500000000000.0d0) then
        tmp = -t / (a / z)
    else if ((x * y) <= 1d+32) then
        tmp = (x * y) * (1.0d0 / a)
    else if ((x * y) <= 5d+93) then
        tmp = -z / (a / t)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e+35) {
		tmp = y / (a / x);
	} else if ((x * y) <= 5e-81) {
		tmp = (z * -t) / a;
	} else if ((x * y) <= 6.6e-23) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 500000000000.0) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 1e+32) {
		tmp = (x * y) * (1.0 / a);
	} else if ((x * y) <= 5e+93) {
		tmp = -z / (a / t);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e+35:
		tmp = y / (a / x)
	elif (x * y) <= 5e-81:
		tmp = (z * -t) / a
	elif (x * y) <= 6.6e-23:
		tmp = (x * y) / a
	elif (x * y) <= 500000000000.0:
		tmp = -t / (a / z)
	elif (x * y) <= 1e+32:
		tmp = (x * y) * (1.0 / a)
	elif (x * y) <= 5e+93:
		tmp = -z / (a / t)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e+35)
		tmp = Float64(y / Float64(a / x));
	elseif (Float64(x * y) <= 5e-81)
		tmp = Float64(Float64(z * Float64(-t)) / a);
	elseif (Float64(x * y) <= 6.6e-23)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 500000000000.0)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (Float64(x * y) <= 1e+32)
		tmp = Float64(Float64(x * y) * Float64(1.0 / a));
	elseif (Float64(x * y) <= 5e+93)
		tmp = Float64(Float64(-z) / Float64(a / t));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e+35)
		tmp = y / (a / x);
	elseif ((x * y) <= 5e-81)
		tmp = (z * -t) / a;
	elseif ((x * y) <= 6.6e-23)
		tmp = (x * y) / a;
	elseif ((x * y) <= 500000000000.0)
		tmp = -t / (a / z);
	elseif ((x * y) <= 1e+32)
		tmp = (x * y) * (1.0 / a);
	elseif ((x * y) <= 5e+93)
		tmp = -z / (a / t);
	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.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+35], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-81], N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6.6e-23], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 500000000000.0], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+32], N[(N[(x * y), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+93], N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (*.f64 x y) < -9.9999999999999997e34

   1. Initial program 89.3%

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

   2. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 80.8%

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

   4. Simplified 80.8%

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

   if -9.9999999999999997e34 < (*.f64 x y) < 4.99999999999999981e-81

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. associate-*r* 81.9%

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

      3. neg-mul-1 81.9%

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

   4. Simplified 81.9%

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

   if 4.99999999999999981e-81 < (*.f64 x y) < 6.60000000000000041e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 80.3%

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

   if 6.60000000000000041e-23 < (*.f64 x y) < 5e11

   1. Initial program 91.9%

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

   2. Taylor expanded in x around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. associate-*r* 65.7%

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

      3. neg-mul-1 65.7%

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

   4. Simplified 65.7%

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

      1. associate-/l* 73.7%

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

      2. distribute-frac-neg 73.7%

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

   6. Applied egg-rr 73.7%

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

   if 5e11 < (*.f64 x y) < 1.00000000000000005e32

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. clear-num 99.1%

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

      2. associate-/r/ 99.7%

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

      3. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.00000000000000005e32 < (*.f64 x y) < 5.0000000000000001e93

   1. Initial program 87.8%

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

      1. div-inv 88.2%

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

      2. fma-neg 88.2%

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

      3. *-commutative 88.2%

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

      4. distribute-rgt-neg-in 88.2%

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

   3. Applied egg-rr 88.2%

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

   4. Taylor expanded in x around 0 87.8%

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

      1. associate-*l/ 75.3%

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

      2. +-commutative 75.3%

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

      3. mul-1-neg 75.3%

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

      4. associate-*r/ 87.3%

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

      5. distribute-lft-neg-in 87.3%

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

      6. cancel-sign-sub-inv 87.3%

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

      7. associate-*l/ 99.8%

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

      8. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around 0 87.8%

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

      1. mul-1-neg 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-/l* 88.0%

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

   9. Simplified 88.0%

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

   if 5.0000000000000001e93 < (*.f64 x y)

   1. Initial program 89.9%

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

   2. Taylor expanded in x around inf 79.8%

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

      1. associate-/l* 89.7%

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

      2. associate-/r/ 87.3%

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

   4. Applied egg-rr 87.3%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]
3. Recombined 7 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \cdot y \leq 6.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 500000000000:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+32}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1.5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 -1.5e+267) (- (/ x (/ a y)) (/ z (/ a t))) (/ t_1 a))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1.5e+267) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if (t_1 <= (-1.5d+267)) then
        tmp = (x / (a / y)) - (z / (a / t))
    else
        tmp = t_1 / a
    end if
    code = tmp
end function

Java

assert x < y;
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 <= -1.5e+267) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -1.5e+267:
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1.5e+267)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -1.5e+267)
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y 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, -1.5e+267], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.5e267

   1. Initial program 76.8%

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

      1. div-sub 71.8%

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

      2. associate-/l* 78.7%

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

      3. associate-/l* 92.3%

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

   3. Applied egg-rr 92.3%

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

   if -1.5e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 97.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1.5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 4: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+262}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) 1e+262) (/ (- (* x y) (* z t)) a) (/ (- t) (/ a z))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 1e+262) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z * t) <= 1d+262) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = -t / (a / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 1e+262) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -t / (a / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= 1e+262:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -t / (a / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= 1e+262)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-t) / Float64(a / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= 1e+262)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -t / (a / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+262], 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 2 regimes

2. if (*.f64 z t) < 1e262

   1. Initial program 96.1%

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

   if 1e262 < (*.f64 z t)

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. associate-*r* 76.2%

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

      3. neg-mul-1 76.2%

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

   4. Simplified 76.2%

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

      1. associate-/l* 99.8%

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

      2. distribute-frac-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+262}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 5: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -11000000000000:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a -11000000000000.0)
   (- (* y (/ x a)) (* t (/ z a)))
   (/ (- (* x y) (* z t)) a)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -11000000000000.0) {
		tmp = (y * (x / a)) - (t * (z / 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.
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 <= (-11000000000000.0d0)) then
        tmp = (y * (x / a)) - (t * (z / a))
    else
        tmp = ((x * y) - (z * t)) / a
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -11000000000000.0) {
		tmp = (y * (x / a)) - (t * (z / a));
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if a <= -11000000000000.0:
		tmp = (y * (x / a)) - (t * (z / a))
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -11000000000000.0)
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / a)));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -11000000000000.0)
		tmp = (y * (x / a)) - (t * (z / 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.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, -11000000000000.0], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / 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 < -1.1e13

   1. Initial program 86.6%

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

      1. div-inv 86.5%

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

      2. fma-neg 86.5%

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

      3. *-commutative 86.5%

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

      4. distribute-rgt-neg-in 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in x around 0 86.6%

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

      1. associate-*l/ 93.2%

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

      2. +-commutative 93.2%

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

      3. mul-1-neg 93.2%

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

      4. associate-*r/ 93.2%

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

      5. distribute-lft-neg-in 93.2%

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

      6. cancel-sign-sub-inv 93.2%

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

      7. associate-*l/ 86.6%

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

      8. associate-*r/ 95.4%

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

   6. Simplified 95.4%

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

   if -1.1e13 < a

   1. Initial program 96.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11000000000000:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 6: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -31500000000000 \lor \neg \left(z \leq 6 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -31500000000000.0) (not (<= z 6e-127)))
   (/ (- t) (/ a z))
   (/ y (/ a x))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -31500000000000.0) || !(z <= 6e-127)) {
		tmp = -t / (a / z);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-31500000000000.0d0)) .or. (.not. (z <= 6d-127))) then
        tmp = -t / (a / z)
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -31500000000000.0) || !(z <= 6e-127)) {
		tmp = -t / (a / z);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (z <= -31500000000000.0) or not (z <= 6e-127):
		tmp = -t / (a / z)
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -31500000000000.0) || !(z <= 6e-127))
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -31500000000000.0) || ~((z <= 6e-127)))
		tmp = -t / (a / z);
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -31500000000000.0], N[Not[LessEqual[z, 6e-127]], $MachinePrecision]], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.15e13 or 6.00000000000000017e-127 < z

   1. Initial program 92.9%

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

   2. Taylor expanded in x around 0 64.2%

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

      1. associate-*r/ 64.2%

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

      2. associate-*r* 64.2%

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

      3. neg-mul-1 64.2%

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

   4. Simplified 64.2%

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

      1. associate-/l* 64.0%

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

      2. distribute-frac-neg 64.0%

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

   6. Applied egg-rr 64.0%

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

   if -3.15e13 < z < 6.00000000000000017e-127

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf 74.6%

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

      1. associate-/l* 72.5%

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

   4. Simplified 72.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -31500000000000 \lor \neg \left(z \leq 6 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 7: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+16)
   (/ (- t) (/ a z))
   (if (<= z 6e-127) (/ y (/ a x)) (* (/ z a) (- t)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+16) {
		tmp = -t / (a / z);
	} else if (z <= 6e-127) {
		tmp = y / (a / x);
	} else {
		tmp = (z / a) * -t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.2d+16)) then
        tmp = -t / (a / z)
    else if (z <= 6d-127) then
        tmp = y / (a / x)
    else
        tmp = (z / a) * -t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+16) {
		tmp = -t / (a / z);
	} else if (z <= 6e-127) {
		tmp = y / (a / x);
	} else {
		tmp = (z / a) * -t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e+16:
		tmp = -t / (a / z)
	elif z <= 6e-127:
		tmp = y / (a / x)
	else:
		tmp = (z / a) * -t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+16)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (z <= 6e-127)
		tmp = Float64(y / Float64(a / x));
	else
		tmp = Float64(Float64(z / a) * Float64(-t));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.2e+16)
		tmp = -t / (a / z);
	elseif (z <= 6e-127)
		tmp = y / (a / x);
	else
		tmp = (z / a) * -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+16], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-127], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e16

   1. Initial program 95.3%

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

   2. Taylor expanded in x around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. associate-*r* 70.4%

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

      3. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

      1. associate-/l* 68.5%

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

      2. distribute-frac-neg 68.5%

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

   6. Applied egg-rr 68.5%

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

   if -5.2e16 < z < 6.00000000000000017e-127

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf 74.6%

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

      1. associate-/l* 72.5%

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

   4. Simplified 72.5%

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

   if 6.00000000000000017e-127 < z

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. associate-*r/ 60.3%

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

      2. mul-1-neg 60.3%

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

      3. distribute-rgt-neg-out 60.3%

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

      4. *-commutative 60.3%

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

      5. associate-/l* 61.9%

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

      6. associate-/r/ 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 8: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{+26}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.48e+26)
   (/ (* z (- t)) a)
   (if (<= z 4.2e-127) (/ y (/ a x)) (* (/ z a) (- t)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.48e+26) {
		tmp = (z * -t) / a;
	} else if (z <= 4.2e-127) {
		tmp = y / (a / x);
	} else {
		tmp = (z / a) * -t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.48d+26)) then
        tmp = (z * -t) / a
    else if (z <= 4.2d-127) then
        tmp = y / (a / x)
    else
        tmp = (z / a) * -t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.48e+26) {
		tmp = (z * -t) / a;
	} else if (z <= 4.2e-127) {
		tmp = y / (a / x);
	} else {
		tmp = (z / a) * -t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.48e+26:
		tmp = (z * -t) / a
	elif z <= 4.2e-127:
		tmp = y / (a / x)
	else:
		tmp = (z / a) * -t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.48e+26)
		tmp = Float64(Float64(z * Float64(-t)) / a);
	elseif (z <= 4.2e-127)
		tmp = Float64(y / Float64(a / x));
	else
		tmp = Float64(Float64(z / a) * Float64(-t));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.48e+26)
		tmp = (z * -t) / a;
	elseif (z <= 4.2e-127)
		tmp = y / (a / x);
	else
		tmp = (z / a) * -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.48e+26], N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 4.2e-127], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.48e26

   1. Initial program 95.1%

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

   2. Taylor expanded in x around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. associate-*r* 72.2%

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

      3. neg-mul-1 72.2%

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

   4. Simplified 72.2%

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

   if -1.48e26 < z < 4.2000000000000002e-127

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 74.5%

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

      1. associate-/l* 72.4%

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

   4. Simplified 72.4%

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

   if 4.2000000000000002e-127 < z

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. associate-*r/ 60.3%

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

      2. mul-1-neg 60.3%

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

      3. distribute-rgt-neg-out 60.3%

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

      4. *-commutative 60.3%

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

      5. associate-/l* 61.9%

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

      6. associate-/r/ 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{+26}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 9: 52.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in x around inf 49.0%

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

   1. associate-*r/ 50.5%

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

4. Simplified 50.5%

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

5. Final simplification 50.5%

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

Alternative 10: 52.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in x around inf 49.0%

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

   1. associate-/l* 50.4%

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

4. Simplified 50.4%

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

5. Final simplification 50.4%

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

Developer target: 91.0% 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 2023230 
(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))