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

Percentage Accurate: 91.2% → 96.9%

Time: 10.4s

Alternatives: 9

Speedup: 0.6×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+267}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+283}:\\ \;\;\;\;\frac{1}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(0 - t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))) (t_2 (fma (/ y a) x (* z (/ t (- 0.0 a))))))
   (if (<= t_1 -5e+267)
     t_2
     (if (<= t_1 1e+283) (/ 1.0 (/ a (fma x y (* z (- 0.0 t))))) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = fma((y / a), x, (z * (t / (0.0 - a))));
	double tmp;
	if (t_1 <= -5e+267) {
		tmp = t_2;
	} else if (t_1 <= 1e+283) {
		tmp = 1.0 / (a / fma(x, y, (z * (0.0 - t))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	t_2 = fma(Float64(y / a), x, Float64(z * Float64(t / Float64(0.0 - a))))
	tmp = 0.0
	if (t_1 <= -5e+267)
		tmp = t_2;
	elseif (t_1 <= 1e+283)
		tmp = Float64(1.0 / Float64(a / fma(x, y, Float64(z * Float64(0.0 - t)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * x + N[(z * N[(t / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+267], t$95$2, If[LessEqual[t$95$1, 1e+283], N[(1.0 / N[(a / N[(x * y + N[(z * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.9999999999999999e267 or 9.99999999999999955e282 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.2%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 86.8

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

   4. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, x, 0 - \frac{z \cdot t}{a}\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. /-lowering-/.f64 93.1

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

   6. Applied egg-rr 93.1%

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

   if -4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999955e282

   1. Initial program 99.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{0 - z \cdot t}\right)}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{1}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{0 - z \cdot t}\right)}} \]

      8. *-lowering-*.f64 99.6

         \[\leadsto \frac{1}{\frac{a}{\mathsf{fma}\left(x, y, 0 - \color{blue}{z \cdot t}\right)}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{fma}\left(x, y, 0 - z \cdot t\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+267}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+283}:\\ \;\;\;\;\frac{1}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(0 - t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+267}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+283}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))) (t_2 (fma (/ y a) x (* z (/ t (- 0.0 a))))))
   (if (<= t_1 -5e+267) t_2 (if (<= t_1 1e+283) (/ t_1 a) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = fma((y / a), x, (z * (t / (0.0 - a))));
	double tmp;
	if (t_1 <= -5e+267) {
		tmp = t_2;
	} else if (t_1 <= 1e+283) {
		tmp = t_1 / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	t_2 = fma(Float64(y / a), x, Float64(z * Float64(t / Float64(0.0 - a))))
	tmp = 0.0
	if (t_1 <= -5e+267)
		tmp = t_2;
	elseif (t_1 <= 1e+283)
		tmp = Float64(t_1 / a);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * x + N[(z * N[(t / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+267], t$95$2, If[LessEqual[t$95$1, 1e+283], N[(t$95$1 / a), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.9999999999999999e267 or 9.99999999999999955e282 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.2%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 86.8

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

   4. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, x, 0 - \frac{z \cdot t}{a}\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. /-lowering-/.f64 93.1

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

   6. Applied egg-rr 93.1%

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

   if -4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999955e282

   1. Initial program 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+267}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+283}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{0 - a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+216}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* y (/ x a))
   (if (<= (* x y) 1e+216) (/ (- (* x y) (* z t)) a) (/ x (/ a y)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e+216) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e+216) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = y * (x / a)
	elif (x * y) <= 1e+216:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = x / (a / y)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 1e+216)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = y * (x / a);
	elseif ((x * y) <= 1e+216)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = x / (a / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+216], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 76.4

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if -inf.0 < (*.f64 x y) < 1e216

   1. Initial program 97.9%

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

   if 1e216 < (*.f64 x y)

   1. Initial program 73.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 73.5

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 96.2

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

   7. Applied egg-rr 96.2%

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

Alternative 4: 92.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{z \cdot \left(0 - t\right)}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 8e-51)
   (* (fma z t (- 0.0 (* x y))) (/ -1.0 a))
   (fma (/ x a) y (/ (* z (- 0.0 t)) a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 8e-51) {
		tmp = fma(z, t, (0.0 - (x * y))) * (-1.0 / a);
	} else {
		tmp = fma((x / a), y, ((z * (0.0 - t)) / a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 8e-51)
		tmp = Float64(fma(z, t, Float64(0.0 - Float64(x * y))) * Float64(-1.0 / a));
	else
		tmp = fma(Float64(x / a), y, Float64(Float64(z * Float64(0.0 - t)) / a));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 8e-51], N[(N[(z * t + N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + N[(N[(z * N[(0.0 - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.0000000000000001e-51

   1. Initial program 95.7%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{0 - x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{0 - x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - \color{blue}{x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{1}{\color{blue}{-1 \cdot a}} \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{a}} \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{\color{blue}{-1}}{a} \]

      15. /-lowering-/.f64 96.2

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \color{blue}{\frac{-1}{a}} \]

   4. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{-1}{a}} \]

   if 8.0000000000000001e-51 < a

   1. Initial program 88.1%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 90.9

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

   4. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, y, 0 - \frac{z \cdot t}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{z \cdot \left(0 - t\right)}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(0 - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+16)
   (* y (/ x a))
   (if (<= (* x y) 1e-85) (/ (* z (- 0.0 t)) a) (* x (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+16) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e-85) {
		tmp = (z * (0.0 - t)) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-5d+16)) then
        tmp = y * (x / a)
    else if ((x * y) <= 1d-85) then
        tmp = (z * (0.0d0 - t)) / a
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+16) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e-85) {
		tmp = (z * (0.0 - t)) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+16:
		tmp = y * (x / a)
	elif (x * y) <= 1e-85:
		tmp = (z * (0.0 - t)) / a
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+16)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 1e-85)
		tmp = Float64(Float64(z * Float64(0.0 - t)) / a);
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+16)
		tmp = y * (x / a);
	elseif ((x * y) <= 1e-85)
		tmp = (z * (0.0 - t)) / a;
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+16], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-85], N[(N[(z * N[(0.0 - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e16

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 80.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 81.7

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

   7. Applied egg-rr 81.7%

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

   if -5e16 < (*.f64 x y) < 9.9999999999999998e-86

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -1 \cdot \frac{t \cdot z}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + 0} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot z}{a}\right)\right)} + 0 \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) + 0 \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(\frac{z}{a}\right)\right)} + 0 \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \frac{z}{a}\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \frac{z}{a}, 0\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(\frac{z}{a}\right)}, 0\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      11. /-lowering-/.f64 80.9

          \[\leadsto \mathsf{fma}\left(t, 0 - \color{blue}{\frac{z}{a}}, 0\right) \]

   5. Simplified 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 0 - \frac{z}{a}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. sub0-neg N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. /-lowering-/.f64 80.6

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

   7. Applied egg-rr 80.6%

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

      1. distribute-rgt-neg-out N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.1

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

   9. Applied egg-rr 87.1%

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

   if 9.9999999999999998e-86 < (*.f64 x y)

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 64.6

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 66.8

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

   7. Applied egg-rr 66.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(0 - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.0002:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;t \cdot \left(0 - \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -0.0002)
   (* y (/ x a))
   (if (<= (* x y) 1e-85) (* t (- 0.0 (/ z a))) (* x (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -0.0002) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e-85) {
		tmp = t * (0.0 - (z / a));
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-0.0002d0)) then
        tmp = y * (x / a)
    else if ((x * y) <= 1d-85) then
        tmp = t * (0.0d0 - (z / a))
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -0.0002) {
		tmp = y * (x / a);
	} else if ((x * y) <= 1e-85) {
		tmp = t * (0.0 - (z / a));
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -0.0002:
		tmp = y * (x / a)
	elif (x * y) <= 1e-85:
		tmp = t * (0.0 - (z / a))
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -0.0002)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 1e-85)
		tmp = Float64(t * Float64(0.0 - Float64(z / a)));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -0.0002)
		tmp = y * (x / a);
	elseif ((x * y) <= 1e-85)
		tmp = t * (0.0 - (z / a));
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -0.0002], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-85], N[(t * N[(0.0 - N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000001e-4

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 79.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 79.2

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

   7. Applied egg-rr 79.2%

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

   if -2.0000000000000001e-4 < (*.f64 x y) < 9.9999999999999998e-86

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -1 \cdot \frac{t \cdot z}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + 0} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot z}{a}\right)\right)} + 0 \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) + 0 \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(\frac{z}{a}\right)\right)} + 0 \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \frac{z}{a}\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \frac{z}{a}, 0\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(\frac{z}{a}\right)}, 0\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      11. /-lowering-/.f64 81.3

          \[\leadsto \mathsf{fma}\left(t, 0 - \color{blue}{\frac{z}{a}}, 0\right) \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 0 - \frac{z}{a}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. sub0-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 81.3

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

   7. Applied egg-rr 81.3%

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

   if 9.9999999999999998e-86 < (*.f64 x y)

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 64.6

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 66.8

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

   7. Applied egg-rr 66.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.0002:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-85}:\\ \;\;\;\;t \cdot \left(0 - \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 1e+216)
   (* (fma z t (- 0.0 (* x y))) (/ -1.0 a))
   (/ x (/ a y))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 1e+216) {
		tmp = fma(z, t, (0.0 - (x * y))) * (-1.0 / a);
	} else {
		tmp = x / (a / y);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 1e+216)
		tmp = Float64(fma(z, t, Float64(0.0 - Float64(x * y))) * Float64(-1.0 / a));
	else
		tmp = Float64(x / Float64(a / y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 1e+216], N[(N[(z * t + N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e216

   1. Initial program 96.0%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{0 - x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{0 - x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - \color{blue}{x \cdot y}\right) \cdot \frac{1}{\mathsf{neg}\left(a\right)} \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{1}{\color{blue}{-1 \cdot a}} \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{a}} \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{\color{blue}{-1}}{a} \]

      15. /-lowering-/.f64 96.3

          \[\leadsto \mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \color{blue}{\frac{-1}{a}} \]

   4. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, 0 - x \cdot y\right) \cdot \frac{-1}{a}} \]

   if 1e216 < (*.f64 x y)

   1. Initial program 73.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 73.5

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 96.2

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

   7. Applied egg-rr 96.2%

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

Alternative 8: 51.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.6e-180) (* x (/ y a)) (* y (/ x a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.6e-180) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 4.6d-180) then
        tmp = x * (y / a)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.6e-180) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.6e-180:
		tmp = x * (y / a)
	else:
		tmp = y * (x / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.6e-180)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 4.6e-180)
		tmp = x * (y / a);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.6e-180], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.59999999999999992e-180

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 56.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 56.2

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

   7. Applied egg-rr 56.2%

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

   if 4.59999999999999992e-180 < t

   1. Initial program 94.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 44.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 44.9

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

   7. Applied egg-rr 44.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. +-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

   3. accelerator-lowering-fma.f64 52.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

5. Simplified 52.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 54.4

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

7. Applied egg-rr 54.4%

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

Developer Target 1: 91.8% accurate, 0.5× 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 2024196 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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