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

Percentage Accurate: 90.9% → 97.4%

Time: 12.0s

Alternatives: 11

Speedup: 0.4×

• 28.1% 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: 90.9% 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: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, z \cdot \frac{t}{-a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 -1e+298)
     (- (* x (/ y a)) (* z (/ t a)))
     (if (<= t_1 1e+288)
       (/ (fma x y (* z (- t))) a)
       (fma y (/ x a) (* z (/ t (- a))))))))

C

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+298) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+288) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = fma(y, (x / a), (z * (t / -a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	elseif (t_1 <= 1e+288)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(y, Float64(x / a), Float64(z * Float64(t / Float64(-a))));
	end
	return tmp
end

Wolfram

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, -1e+298], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+288], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision] + N[(z * N[(t / (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999996e297

   1. Initial program 65.8%

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

      1. div-sub 62.8%

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

      2. associate-/l* 77.1%

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

      3. associate-/l* 96.8%

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

   4. Applied egg-rr 96.8%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e288

   1. Initial program 98.1%

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

      1. div-sub 97.0%

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

      2. *-commutative 97.0%

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

      3. div-sub 98.1%

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

      4. *-commutative 98.1%

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

      5. fma-neg 98.1%

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

      6. distribute-rgt-neg-out 98.1%

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

   3. Simplified 98.1%

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

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

   1. Initial program 64.8%

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

      1. div-sub 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-/l* 64.6%

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

      4. fma-neg 69.8%

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

      5. associate-/l* 97.2%

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

   4. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{a}, -z \cdot \frac{t}{a}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, z \cdot \frac{t}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 -1e+298)
     (- (* x (/ y a)) (* z (/ t a)))
     (if (<= t_1 1e+288)
       (/ (fma x y (* z (- t))) a)
       (- (* y (/ x a)) (* t (/ z a)))))))

C

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+298) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+288) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	elseif (t_1 <= 1e+288)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / a)));
	end
	return tmp
end

Wolfram

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, -1e+298], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+288], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999996e297

   1. Initial program 65.8%

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

      1. div-sub 62.8%

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

      2. associate-/l* 77.1%

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

      3. associate-/l* 96.8%

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

   4. Applied egg-rr 96.8%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e288

   1. Initial program 98.1%

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

      1. div-sub 97.0%

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

      2. *-commutative 97.0%

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

      3. div-sub 98.1%

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

      4. *-commutative 98.1%

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

      5. fma-neg 98.1%

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

      6. distribute-rgt-neg-out 98.1%

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

   3. Simplified 98.1%

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

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

   1. Initial program 64.8%

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

      1. div-sub 59.5%

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

      2. *-un-lft-identity 59.5%

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

      3. add-sqr-sqrt 32.2%

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

      4. times-frac 32.2%

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

      5. fma-neg 32.2%

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

      6. associate-/l* 42.2%

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

   4. Applied egg-rr 42.2%

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

      1. fma-undefine 42.2%

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

      2. distribute-lft-neg-in 42.2%

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

      3. cancel-sign-sub-inv 42.2%

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

      4. associate-*l/ 42.2%

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

      5. *-lft-identity 42.2%

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

      6. *-commutative 42.2%

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

      7. associate-/l* 44.7%

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

      8. associate-*r/ 34.7%

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

      9. *-commutative 34.7%

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

      10. associate-/l* 44.7%

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

   6. Simplified 44.7%

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

      1. associate-/l* 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-/l/ 44.7%

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

      4. add-sqr-sqrt 92.0%

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

   8. Applied egg-rr 92.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298} \lor \neg \left(t\_1 \leq 10^{+288}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 -1e+298) (not (<= t_1 1e+288)))
     (- (* x (/ y a)) (* z (/ t a)))
     (/ t_1 a))))

C

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+298) || !(t_1 <= 1e+288)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else {
		tmp = t_1 / a;
	}
	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 = (x * y) - (z * t)
    if ((t_1 <= (-1d+298)) .or. (.not. (t_1 <= 1d+288))) then
        tmp = (x * (y / a)) - (z * (t / a))
    else
        tmp = t_1 / a
    end if
    code = tmp
end function

Java

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+298) || !(t_1 <= 1e+288)) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -1e+298) or not (t_1 <= 1e+288):
		tmp = (x * (y / a)) - (z * (t / a))
	else:
		tmp = t_1 / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -1e+298) || !(t_1 <= 1e+288))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -1e+298) || ~((t_1 <= 1e+288)))
		tmp = (x * (y / a)) - (z * (t / a));
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

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+298], N[Not[LessEqual[t$95$1, 1e+288]], $MachinePrecision]], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $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)) < -9.9999999999999996e297 or 1e288 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 65.3%

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

      1. div-sub 61.0%

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

      2. associate-/l* 70.4%

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

      3. associate-/l* 94.2%

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

   4. Applied egg-rr 94.2%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e288

   1. Initial program 98.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+298} \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+288}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+288}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 -1e+298)
     (- (* x (/ y a)) (* z (/ t a)))
     (if (<= t_1 1e+288) (/ t_1 a) (- (* y (/ x a)) (* t (/ z a)))))))

C

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+298) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+288) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / a));
	}
	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 = (x * y) - (z * t)
    if (t_1 <= (-1d+298)) then
        tmp = (x * (y / a)) - (z * (t / a))
    else if (t_1 <= 1d+288) then
        tmp = t_1 / a
    else
        tmp = (y * (x / a)) - (t * (z / a))
    end if
    code = tmp
end function

Java

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+298) {
		tmp = (x * (y / a)) - (z * (t / a));
	} else if (t_1 <= 1e+288) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -1e+298:
		tmp = (x * (y / a)) - (z * (t / a))
	elif t_1 <= 1e+288:
		tmp = t_1 / a
	else:
		tmp = (y * (x / a)) - (t * (z / a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)));
	elseif (t_1 <= 1e+288)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -1e+298)
		tmp = (x * (y / a)) - (z * (t / a));
	elseif (t_1 <= 1e+288)
		tmp = t_1 / a;
	else
		tmp = (y * (x / a)) - (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

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, -1e+298], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+288], N[(t$95$1 / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999996e297

   1. Initial program 65.8%

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

      1. div-sub 62.8%

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

      2. associate-/l* 77.1%

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

      3. associate-/l* 96.8%

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

   4. Applied egg-rr 96.8%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e288

   1. Initial program 98.1%

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

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

   1. Initial program 64.8%

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

      1. div-sub 59.5%

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

      2. *-un-lft-identity 59.5%

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

      3. add-sqr-sqrt 32.2%

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

      4. times-frac 32.2%

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

      5. fma-neg 32.2%

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

      6. associate-/l* 42.2%

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

   4. Applied egg-rr 42.2%

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

      1. fma-undefine 42.2%

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

      2. distribute-lft-neg-in 42.2%

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

      3. cancel-sign-sub-inv 42.2%

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

      4. associate-*l/ 42.2%

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

      5. *-lft-identity 42.2%

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

      6. *-commutative 42.2%

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

      7. associate-/l* 44.7%

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

      8. associate-*r/ 34.7%

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

      9. *-commutative 34.7%

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

      10. associate-/l* 44.7%

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

   6. Simplified 44.7%

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

      1. associate-/l* 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-/l/ 44.7%

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

      4. add-sqr-sqrt 92.0%

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

   8. Applied egg-rr 92.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+298}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+288}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{z}{-\frac{a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+305}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) -2e+282)
   (/ z (- (/ a t)))
   (if (<= (* z t) 1e+305) (/ (- (* x y) (* z t)) a) (/ t (/ a (- z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -2e+282) {
		tmp = z / -(a / t);
	} else if ((z * t) <= 1e+305) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t / (a / -z);
	}
	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) :: tmp
    if ((z * t) <= (-2d+282)) then
        tmp = z / -(a / t)
    else if ((z * t) <= 1d+305) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t / (a / -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -2e+282) {
		tmp = z / -(a / t);
	} else if ((z * t) <= 1e+305) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t / (a / -z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -2e+282:
		tmp = z / -(a / t)
	elif (z * t) <= 1e+305:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t / (a / -z)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= -2e+282)
		tmp = Float64(z / Float64(-Float64(a / t)));
	elseif (Float64(z * t) <= 1e+305)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(t / Float64(a / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -2e+282)
		tmp = z / -(a / t);
	elseif ((z * t) <= 1e+305)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t / (a / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+282], N[(z / (-N[(a / t), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+305], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(t / N[(a / (-z)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000007e282

   1. Initial program 63.0%

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

      1. div-sub 56.5%

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

      2. *-un-lft-identity 56.5%

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

      3. add-sqr-sqrt 32.8%

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

      4. times-frac 32.8%

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

      5. fma-neg 32.8%

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

      6. associate-/l* 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. fma-undefine 45.1%

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

      2. distribute-lft-neg-in 45.1%

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

      3. cancel-sign-sub-inv 45.1%

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

      4. associate-*l/ 45.1%

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

      5. *-lft-identity 45.1%

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

      6. *-commutative 45.1%

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

      7. associate-/l* 45.1%

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

      8. associate-*r/ 32.8%

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

      9. *-commutative 32.8%

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

      10. associate-/l* 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in y around 0 66.2%

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

      1. mul-1-neg 66.2%

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

      2. associate-*r/ 93.7%

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

      3. distribute-rgt-neg-in 93.7%

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

      4. distribute-neg-frac 93.7%

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

   9. Simplified 93.7%

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

       1. distribute-frac-neg 93.7%

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

       2. distribute-rgt-neg-in 93.7%

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

       3. *-commutative 93.7%

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

       4. associate-/r/ 93.6%

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

       5. distribute-neg-frac2 93.6%

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

       6. distribute-neg-frac2 93.6%

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

   11. Applied egg-rr 93.6%

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

   if -2.00000000000000007e282 < (*.f64 z t) < 9.9999999999999994e304

   1. Initial program 95.2%

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

   if 9.9999999999999994e304 < (*.f64 z t)

   1. Initial program 53.4%

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

      1. div-sub 53.4%

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

      2. *-un-lft-identity 53.4%

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

      3. add-sqr-sqrt 15.2%

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

      4. times-frac 15.2%

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

      5. fma-neg 15.2%

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

      6. associate-/l* 28.5%

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

   4. Applied egg-rr 28.5%

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

      1. fma-undefine 28.5%

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

      2. distribute-lft-neg-in 28.5%

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

      3. cancel-sign-sub-inv 28.5%

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

      4. associate-*l/ 28.5%

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

      5. *-lft-identity 28.5%

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

      6. *-commutative 28.5%

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

      7. associate-/l* 28.5%

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

      8. associate-*r/ 15.2%

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

      9. *-commutative 15.2%

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

      10. associate-/l* 28.5%

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

   6. Simplified 28.5%

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

   7. Taylor expanded in y around 0 53.4%

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

      1. mul-1-neg 53.4%

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

      2. associate-*r/ 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

      4. distribute-neg-frac 99.4%

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

   9. Simplified 99.4%

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

       1. distribute-frac-neg 99.4%

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

       2. distribute-rgt-neg-in 99.4%

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

       3. distribute-lft-neg-in 99.4%

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

       4. clear-num 99.6%

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

       5. un-div-inv 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+282}:\\ \;\;\;\;\frac{z}{-\frac{a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+305}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e-11)
   (* y (/ x a))
   (if (<= (* x y) 5e-108) (* t (/ z (- a))) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e-11) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e-108) {
		tmp = t * (z / -a);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if ((x * y) <= (-1d-11)) then
        tmp = y * (x / a)
    else if ((x * y) <= 5d-108) then
        tmp = t * (z / -a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e-11) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e-108) {
		tmp = t * (z / -a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e-11:
		tmp = y * (x / a)
	elif (x * y) <= 5e-108:
		tmp = t * (z / -a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e-11)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 5e-108)
		tmp = Float64(t * Float64(z / Float64(-a)));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e-11)
		tmp = y * (x / a);
	elseif ((x * y) <= 5e-108)
		tmp = t * (z / -a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e-11], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-108], N[(t * N[(z / (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999939e-12

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*r/ 76.2%

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

   5. Simplified 76.2%

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

   if -9.99999999999999939e-12 < (*.f64 x y) < 5e-108

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. associate-/l* 86.4%

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

      3. distribute-rgt-neg-in 86.4%

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

   5. Simplified 86.4%

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

   if 5e-108 < (*.f64 x y)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf 69.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -1e-11)
   (* y (/ x a))
   (if (<= (* x y) 5e-108) (/ t (/ a (- z))) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e-11) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e-108) {
		tmp = t / (a / -z);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if ((x * y) <= (-1d-11)) then
        tmp = y * (x / a)
    else if ((x * y) <= 5d-108) then
        tmp = t / (a / -z)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -1e-11) {
		tmp = y * (x / a);
	} else if ((x * y) <= 5e-108) {
		tmp = t / (a / -z);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -1e-11:
		tmp = y * (x / a)
	elif (x * y) <= 5e-108:
		tmp = t / (a / -z)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -1e-11)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 5e-108)
		tmp = Float64(t / Float64(a / Float64(-z)));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -1e-11)
		tmp = y * (x / a);
	elseif ((x * y) <= 5e-108)
		tmp = t / (a / -z);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e-11], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-108], N[(t / N[(a / (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999939e-12

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*r/ 76.2%

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

   5. Simplified 76.2%

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

   if -9.99999999999999939e-12 < (*.f64 x y) < 5e-108

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. *-un-lft-identity 90.3%

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

      3. add-sqr-sqrt 47.5%

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

      4. times-frac 47.5%

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

      5. fma-neg 47.5%

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

      6. associate-/l* 43.1%

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

   4. Applied egg-rr 43.1%

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

      1. fma-undefine 43.1%

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

      2. distribute-lft-neg-in 43.1%

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

      3. cancel-sign-sub-inv 43.1%

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

      4. associate-*l/ 43.1%

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

      5. *-lft-identity 43.1%

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

      6. *-commutative 43.1%

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

      7. associate-/l* 44.1%

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

      8. associate-*r/ 48.4%

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

      9. *-commutative 48.4%

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

      10. associate-/l* 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in y around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. associate-*r/ 86.4%

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

      3. distribute-rgt-neg-in 86.4%

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

      4. distribute-neg-frac 86.4%

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

   9. Simplified 86.4%

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

       1. distribute-frac-neg 86.4%

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

       2. distribute-rgt-neg-in 86.4%

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

       3. distribute-lft-neg-in 86.4%

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

       4. clear-num 86.5%

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

       5. un-div-inv 86.6%

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

   11. Applied egg-rr 86.6%

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

   if 5e-108 < (*.f64 x y)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf 69.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -3.8e-115) (* y (/ x a)) (* x (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -3.8e-115) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	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) :: tmp
    if (x <= (-3.8d-115)) then
        tmp = y * (x / a)
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -3.8e-115) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -3.8e-115:
		tmp = y * (x / a)
	else:
		tmp = x * (y / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -3.8e-115)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -3.8e-115)
		tmp = y * (x / a);
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -3.8e-115], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999992e-115

   1. Initial program 88.2%

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

   3. Taylor expanded in x around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*r/ 68.3%

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

   5. Simplified 68.3%

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

   if -3.79999999999999992e-115 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf 47.2%

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

      1. associate-*r/ 45.1%

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

   5. Simplified 45.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-245}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.75e-245) (* y (/ x a)) (/ x (/ a y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.75e-245) {
		tmp = y * (x / a);
	} else {
		tmp = x / (a / y);
	}
	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) :: tmp
    if (x <= (-1.75d-245)) then
        tmp = y * (x / a)
    else
        tmp = x / (a / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.75e-245) {
		tmp = y * (x / a);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.75e-245:
		tmp = y * (x / a)
	else:
		tmp = x / (a / y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.75e-245)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.75e-245)
		tmp = y * (x / a);
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.75e-245], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75000000000000008e-245

   1. Initial program 88.3%

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

   3. Taylor expanded in x around inf 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*r/ 54.8%

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

   5. Simplified 54.8%

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

   if -1.75000000000000008e-245 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf 52.0%

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

      1. associate-*r/ 49.5%

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

   5. Simplified 49.5%

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

      1. clear-num 48.8%

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

      2. un-div-inv 48.6%

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

   7. Applied egg-rr 48.6%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-245}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.35e-127) (* y (/ x a)) (/ (* x y) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.35e-127) {
		tmp = y * (x / a);
	} else {
		tmp = (x * y) / a;
	}
	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) :: tmp
    if (x <= (-1.35d-127)) then
        tmp = y * (x / a)
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.35e-127) {
		tmp = y * (x / a);
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.35e-127:
		tmp = y * (x / a)
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.35e-127)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.35e-127)
		tmp = y * (x / a);
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.35e-127], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e-127

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*r/ 65.7%

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

   5. Simplified 65.7%

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

   if -1.35e-127 < x

   1. Initial program 89.1%

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

   3. Taylor expanded in x around inf 47.9%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x * (y / 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 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

3. Taylor expanded in x around inf 51.8%

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

   1. associate-*r/ 51.5%

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

5. Simplified 51.5%

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

6. Final simplification 51.5%

   \[\leadsto x \cdot \frac{y}{a} \]
7. Add Preprocessing

Developer target: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024043 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))