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

Percentage Accurate: 91.5% → 96.7%

Time: 7.5s

Alternatives: 8

Speedup: 0.7×

• 29.5% 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: 91.5% 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: 96.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+229}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (if (<= t_1 4e+229) (/ t_1 a) (- (/ x (/ a y)) (/ z (/ a t)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else if (t_1 <= 4e+229) {
		tmp = t_1 / a;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	elseif (t_1 <= 4e+229)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+229], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.2%

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

   2. Taylor expanded in x around 0 46.5%

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

      1. fma-def 46.5%

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

      2. associate-/l* 86.9%

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

      3. associate-/l* 91.1%

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

   4. Simplified 91.1%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4e229

   1. Initial program 97.6%

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

   if 4e229 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 84.1%

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

      1. div-sub 84.1%

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

      2. associate-/l* 90.5%

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

      3. associate-/l* 98.2%

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

   3. Applied egg-rr 98.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 4 \cdot 10^{+229}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+281} \lor \neg \left(t_1 \leq 4 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 -1e+281) (not (<= t_1 4e+229)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ t_1 a))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -1e+281) || !(t_1 <= 4e+229)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if ((t_1 <= (-1d+281)) .or. (.not. (t_1 <= 4d+229))) then
        tmp = (x / (a / y)) - (z / (a / t))
    else
        tmp = t_1 / a
    end if
    code = tmp
end function

Java

assert z < t;
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 <= -1e+281) || !(t_1 <= 4e+229)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -1e+281) or not (t_1 <= 4e+229):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / a
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -1e+281) || !(t_1 <= 4e+229))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -1e+281) || ~((t_1 <= 4e+229)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+281], N[Not[LessEqual[t$95$1, 4e+229]], $MachinePrecision]], 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)) < -1e281 or 4e229 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 77.4%

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

      1. div-sub 75.2%

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

      2. associate-/l* 80.6%

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

      3. associate-/l* 95.3%

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

   3. Applied egg-rr 95.3%

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

   if -1e281 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4e229

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+281} \lor \neg \left(x \cdot y - z \cdot t \leq 4 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 5000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (/ a y))))
   (if (<= (* z t) -5e+51)
     (* t (- (/ z a)))
     (if (<= (* z t) -5e+26)
       t_1
       (if (<= (* z t) -5e-12)
         (/ (- (* z t)) a)
         (if (<= (* z t) 5000000.0) t_1 (- (* (/ t a) z))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (a / y);
	double tmp;
	if ((z * t) <= -5e+51) {
		tmp = t * -(z / a);
	} else if ((z * t) <= -5e+26) {
		tmp = t_1;
	} else if ((z * t) <= -5e-12) {
		tmp = -(z * t) / a;
	} else if ((z * t) <= 5000000.0) {
		tmp = t_1;
	} else {
		tmp = -((t / a) * z);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (a / y)
    if ((z * t) <= (-5d+51)) then
        tmp = t * -(z / a)
    else if ((z * t) <= (-5d+26)) then
        tmp = t_1
    else if ((z * t) <= (-5d-12)) then
        tmp = -(z * t) / a
    else if ((z * t) <= 5000000.0d0) then
        tmp = t_1
    else
        tmp = -((t / a) * z)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (a / y);
	double tmp;
	if ((z * t) <= -5e+51) {
		tmp = t * -(z / a);
	} else if ((z * t) <= -5e+26) {
		tmp = t_1;
	} else if ((z * t) <= -5e-12) {
		tmp = -(z * t) / a;
	} else if ((z * t) <= 5000000.0) {
		tmp = t_1;
	} else {
		tmp = -((t / a) * z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = x / (a / y)
	tmp = 0
	if (z * t) <= -5e+51:
		tmp = t * -(z / a)
	elif (z * t) <= -5e+26:
		tmp = t_1
	elif (z * t) <= -5e-12:
		tmp = -(z * t) / a
	elif (z * t) <= 5000000.0:
		tmp = t_1
	else:
		tmp = -((t / a) * z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(a / y))
	tmp = 0.0
	if (Float64(z * t) <= -5e+51)
		tmp = Float64(t * Float64(-Float64(z / a)));
	elseif (Float64(z * t) <= -5e+26)
		tmp = t_1;
	elseif (Float64(z * t) <= -5e-12)
		tmp = Float64(Float64(-Float64(z * t)) / a);
	elseif (Float64(z * t) <= 5000000.0)
		tmp = t_1;
	else
		tmp = Float64(-Float64(Float64(t / a) * z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (a / y);
	tmp = 0.0;
	if ((z * t) <= -5e+51)
		tmp = t * -(z / a);
	elseif ((z * t) <= -5e+26)
		tmp = t_1;
	elseif ((z * t) <= -5e-12)
		tmp = -(z * t) / a;
	elseif ((z * t) <= 5000000.0)
		tmp = t_1;
	else
		tmp = -((t / a) * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+51], N[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -5e+26], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -5e-12], N[((-N[(z * t), $MachinePrecision]) / a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5000000.0], t$95$1, (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision])]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -5e51

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. associate-*r/ 81.3%

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

      2. mul-1-neg 81.3%

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

      3. distribute-rgt-neg-out 81.3%

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

      4. *-commutative 81.3%

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

      5. associate-/l* 84.7%

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

      6. associate-/r/ 84.2%

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

   4. Simplified 84.2%

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

   if -5e51 < (*.f64 z t) < -5.0000000000000001e26 or -4.9999999999999997e-12 < (*.f64 z t) < 5e6

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 78.0%

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

      1. associate-*r/ 76.6%

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

   4. Simplified 76.6%

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

      1. associate-*r/ 78.0%

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

      2. *-commutative 78.0%

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

      3. associate-*r/ 80.1%

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

      4. clear-num 80.0%

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

      5. un-div-inv 80.7%

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

   6. Applied egg-rr 80.7%

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

   if -5.0000000000000001e26 < (*.f64 z t) < -4.9999999999999997e-12

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. associate-*r* 66.5%

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

      3. neg-mul-1 66.5%

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

   4. Simplified 66.5%

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

   if 5e6 < (*.f64 z t)

   1. Initial program 80.3%

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

      1. div-sub 78.4%

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

      2. *-un-lft-identity 78.4%

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

      3. times-frac 74.8%

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

      4. fma-neg 74.8%

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

      5. associate-/l* 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in x around 0 64.7%

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

      1. mul-1-neg 64.7%

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

      2. associate-*r/ 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-*l/ 64.7%

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

      5. associate-*r/ 80.5%

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

      6. distribute-rgt-neg-out 80.5%

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

      7. distribute-neg-frac 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 5000000:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \end{array} \]

Alternative 4: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x y) a)))
   (if (<= x -3e+84)
     t_1
     (if (<= x -3.7e-66)
       (* t (- (/ z a)))
       (if (<= x -1.5e-115)
         t_1
         (if (<= x 1.22e-116) (- (* (/ t a) z)) (* y (/ x a))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if (x <= -3e+84) {
		tmp = t_1;
	} else if (x <= -3.7e-66) {
		tmp = t * -(z / a);
	} else if (x <= -1.5e-115) {
		tmp = t_1;
	} else if (x <= 1.22e-116) {
		tmp = -((t / a) * z);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) / a
    if (x <= (-3d+84)) then
        tmp = t_1
    else if (x <= (-3.7d-66)) then
        tmp = t * -(z / a)
    else if (x <= (-1.5d-115)) then
        tmp = t_1
    else if (x <= 1.22d-116) then
        tmp = -((t / a) * z)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if (x <= -3e+84) {
		tmp = t_1;
	} else if (x <= -3.7e-66) {
		tmp = t * -(z / a);
	} else if (x <= -1.5e-115) {
		tmp = t_1;
	} else if (x <= 1.22e-116) {
		tmp = -((t / a) * z);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) / a
	tmp = 0
	if x <= -3e+84:
		tmp = t_1
	elif x <= -3.7e-66:
		tmp = t * -(z / a)
	elif x <= -1.5e-115:
		tmp = t_1
	elif x <= 1.22e-116:
		tmp = -((t / a) * z)
	else:
		tmp = y * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / a)
	tmp = 0.0
	if (x <= -3e+84)
		tmp = t_1;
	elseif (x <= -3.7e-66)
		tmp = Float64(t * Float64(-Float64(z / a)));
	elseif (x <= -1.5e-115)
		tmp = t_1;
	elseif (x <= 1.22e-116)
		tmp = Float64(-Float64(Float64(t / a) * z));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) / a;
	tmp = 0.0;
	if (x <= -3e+84)
		tmp = t_1;
	elseif (x <= -3.7e-66)
		tmp = t * -(z / a);
	elseif (x <= -1.5e-115)
		tmp = t_1;
	elseif (x <= 1.22e-116)
		tmp = -((t / a) * z);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[x, -3e+84], t$95$1, If[LessEqual[x, -3.7e-66], N[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, -1.5e-115], t$95$1, If[LessEqual[x, 1.22e-116], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.99999999999999996e84 or -3.7000000000000002e-66 < x < -1.5000000000000001e-115

   1. Initial program 91.7%

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

   2. Taylor expanded in x around inf 74.4%

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

   if -2.99999999999999996e84 < x < -3.7000000000000002e-66

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

      3. distribute-rgt-neg-out 57.3%

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

      4. *-commutative 57.3%

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

      5. associate-/l* 60.4%

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

      6. associate-/r/ 60.5%

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

   4. Simplified 60.5%

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

   if -1.5000000000000001e-115 < x < 1.22e-116

   1. Initial program 90.7%

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

      1. div-sub 90.7%

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

      2. *-un-lft-identity 90.7%

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

      3. times-frac 83.7%

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

      4. fma-neg 83.7%

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

      5. associate-/l* 86.4%

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

   3. Applied egg-rr 86.4%

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

   4. Taylor expanded in x around 0 73.0%

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

      1. mul-1-neg 73.0%

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

      2. associate-*r/ 72.9%

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

      3. *-commutative 72.9%

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

      4. associate-*l/ 73.0%

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

      5. associate-*r/ 72.0%

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

      6. distribute-rgt-neg-out 72.0%

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

      7. distribute-neg-frac 72.0%

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

   6. Simplified 72.0%

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

   if 1.22e-116 < x

   1. Initial program 89.3%

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

   2. Taylor expanded in x around inf 61.4%

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

      1. associate-*r/ 66.0%

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

   4. Simplified 66.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+84}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 5: 63.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{z}{-\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x y) a)))
   (if (<= x -6.2e+84)
     t_1
     (if (<= x -6.2e-66)
       (* t (- (/ z a)))
       (if (<= x -1.6e-115)
         t_1
         (if (<= x 2e-118) (/ z (- (/ a t))) (* y (/ x a))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if (x <= -6.2e+84) {
		tmp = t_1;
	} else if (x <= -6.2e-66) {
		tmp = t * -(z / a);
	} else if (x <= -1.6e-115) {
		tmp = t_1;
	} else if (x <= 2e-118) {
		tmp = z / -(a / t);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) / a
    if (x <= (-6.2d+84)) then
        tmp = t_1
    else if (x <= (-6.2d-66)) then
        tmp = t * -(z / a)
    else if (x <= (-1.6d-115)) then
        tmp = t_1
    else if (x <= 2d-118) then
        tmp = z / -(a / t)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if (x <= -6.2e+84) {
		tmp = t_1;
	} else if (x <= -6.2e-66) {
		tmp = t * -(z / a);
	} else if (x <= -1.6e-115) {
		tmp = t_1;
	} else if (x <= 2e-118) {
		tmp = z / -(a / t);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) / a
	tmp = 0
	if x <= -6.2e+84:
		tmp = t_1
	elif x <= -6.2e-66:
		tmp = t * -(z / a)
	elif x <= -1.6e-115:
		tmp = t_1
	elif x <= 2e-118:
		tmp = z / -(a / t)
	else:
		tmp = y * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / a)
	tmp = 0.0
	if (x <= -6.2e+84)
		tmp = t_1;
	elseif (x <= -6.2e-66)
		tmp = Float64(t * Float64(-Float64(z / a)));
	elseif (x <= -1.6e-115)
		tmp = t_1;
	elseif (x <= 2e-118)
		tmp = Float64(z / Float64(-Float64(a / t)));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) / a;
	tmp = 0.0;
	if (x <= -6.2e+84)
		tmp = t_1;
	elseif (x <= -6.2e-66)
		tmp = t * -(z / a);
	elseif (x <= -1.6e-115)
		tmp = t_1;
	elseif (x <= 2e-118)
		tmp = z / -(a / t);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[x, -6.2e+84], t$95$1, If[LessEqual[x, -6.2e-66], N[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, -1.6e-115], t$95$1, If[LessEqual[x, 2e-118], N[(z / (-N[(a / t), $MachinePrecision])), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.20000000000000006e84 or -6.1999999999999995e-66 < x < -1.6e-115

   1. Initial program 91.7%

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

   2. Taylor expanded in x around inf 74.4%

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

   if -6.20000000000000006e84 < x < -6.1999999999999995e-66

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

      3. distribute-rgt-neg-out 57.3%

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

      4. *-commutative 57.3%

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

      5. associate-/l* 60.4%

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

      6. associate-/r/ 60.5%

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

   4. Simplified 60.5%

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

   if -1.6e-115 < x < 1.99999999999999997e-118

   1. Initial program 90.7%

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

   2. Taylor expanded in x around 0 73.0%

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

      1. associate-*r/ 73.0%

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

      2. mul-1-neg 73.0%

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

      3. distribute-rgt-neg-out 73.0%

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

      4. *-commutative 73.0%

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

      5. associate-/l* 73.8%

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

      6. associate-/r/ 72.9%

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

   4. Simplified 72.9%

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

      1. distribute-frac-neg 72.9%

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

      2. distribute-lft-neg-in 72.9%

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

      3. associate-/r/ 73.8%

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

      4. frac-2neg 73.8%

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

      5. distribute-neg-frac 73.8%

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

      6. remove-double-neg 73.8%

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

      7. distribute-neg-frac 73.8%

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

   6. Applied egg-rr 73.8%

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

   if 1.99999999999999997e-118 < x

   1. Initial program 89.3%

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

   2. Taylor expanded in x around inf 61.4%

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

      1. associate-*r/ 66.0%

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

   4. Simplified 66.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{z}{-\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 6: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 4 \cdot 10^{+254}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) 4e+254) (/ (- (* x y) (* z t)) a) (- (* (/ t a) z))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 4e+254) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -((t / a) * z);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z * t) <= 4d+254) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = -((t / a) * z)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 4e+254) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -((t / a) * z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= 4e+254:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -((t / a) * z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= 4e+254)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(-Float64(Float64(t / a) * z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= 4e+254)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -((t / a) * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], 4e+254], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 3.9999999999999997e254

   1. Initial program 94.1%

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

   if 3.9999999999999997e254 < (*.f64 z t)

   1. Initial program 53.0%

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

      1. div-sub 48.5%

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

      2. *-un-lft-identity 48.5%

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

      3. times-frac 48.5%

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

      4. fma-neg 48.5%

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

      5. associate-/l* 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in x around 0 53.0%

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

      1. mul-1-neg 53.0%

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

      2. associate-*r/ 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*l/ 53.0%

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

      5. associate-*r/ 95.3%

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

      6. distribute-rgt-neg-out 95.3%

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

      7. distribute-neg-frac 95.3%

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

   6. Simplified 95.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 4 \cdot 10^{+254}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \end{array} \]

Alternative 7: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.2e-104)
   (/ (* x y) a)
   (if (<= y 2.85e+38) (- (* (/ t a) z)) (* x (/ y a)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.2e-104) {
		tmp = (x * y) / a;
	} else if (y <= 2.85e+38) {
		tmp = -((t / a) * z);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-2.2d-104)) then
        tmp = (x * y) / a
    else if (y <= 2.85d+38) then
        tmp = -((t / a) * z)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.2e-104) {
		tmp = (x * y) / a;
	} else if (y <= 2.85e+38) {
		tmp = -((t / a) * z);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.2e-104:
		tmp = (x * y) / a
	elif y <= 2.85e+38:
		tmp = -((t / a) * z)
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.2e-104)
		tmp = Float64(Float64(x * y) / a);
	elseif (y <= 2.85e+38)
		tmp = Float64(-Float64(Float64(t / a) * z));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2.2e-104)
		tmp = (x * y) / a;
	elseif (y <= 2.85e+38)
		tmp = -((t / a) * z);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.2e-104], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 2.85e+38], (-N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000012e-104

   1. Initial program 95.0%

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

   2. Taylor expanded in x around inf 55.8%

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

   if -2.20000000000000012e-104 < y < 2.8499999999999999e38

   1. Initial program 92.5%

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

      1. div-sub 92.5%

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

      2. *-un-lft-identity 92.5%

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

      3. times-frac 92.5%

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

      4. fma-neg 92.5%

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

      5. associate-/l* 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in x around 0 65.9%

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

      1. mul-1-neg 65.9%

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

      2. associate-*r/ 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-*l/ 65.9%

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

      5. associate-*r/ 69.0%

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

      6. distribute-rgt-neg-out 69.0%

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

      7. distribute-neg-frac 69.0%

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

   6. Simplified 69.0%

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

   if 2.8499999999999999e38 < y

   1. Initial program 81.3%

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

   2. Taylor expanded in x around inf 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*r/ 77.0%

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

   4. Applied egg-rr 77.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8: 51.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in x around inf 49.6%

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

   1. *-commutative 49.6%

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

   2. associate-*r/ 53.0%

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

4. Applied egg-rr 53.0%

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

5. Final simplification 53.0%

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

Developer target: 91.7% 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 2023221 
(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))