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

Percentage Accurate: 91.6% → 97.4%

Time: 9.8s

Alternatives: 10

Speedup: 0.6×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(\frac{y}{a}, x, -z \cdot \frac{t}{a}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 a))))))
   (if (<= t_1 -2e+278)
     t_2
     (if (<= t_1 4e+292) (/ (fma y x (* z (- t))) 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 / a)));
	double tmp;
	if (t_1 <= -2e+278) {
		tmp = t_2;
	} else if (t_1 <= 4e+292) {
		tmp = fma(y, x, (z * -t)) / 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(-Float64(z * Float64(t / a))))
	tmp = 0.0
	if (t_1 <= -2e+278)
		tmp = t_2;
	elseif (t_1 <= 4e+292)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / 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 / a), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+278], t$95$2, If[LessEqual[t$95$1, 4e+292], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.99999999999999993e278 or 4.0000000000000001e292 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.6%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 65.2

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

   4. Applied egg-rr 65.2%

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

      1. unsub-neg N/A

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

      2. *-commutative N/A

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

      3. sub-div N/A

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

      4. associate-*l/ N/A

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

      5. unsub-neg N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. distribute-neg-frac N/A

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

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

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

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

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

      14. /-lowering-/.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

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

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

      19. distribute-frac-neg N/A

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

      20. distribute-frac-neg2 N/A

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

      21. /-lowering-/.f64 N/A

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

      22. neg-lowering-neg.f64 91.7

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

   6. Applied egg-rr 91.7%

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

   if -1.99999999999999993e278 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.0000000000000001e292

   1. Initial program 98.8%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.8

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

   4. Applied egg-rr 98.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, -z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, -z \cdot \frac{t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+298}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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
 (let* ((t_1 (/ (- (* x y) (* z t)) a)))
   (if (<= t_1 -2e+169)
     (* x (/ y a))
     (if (<= t_1 1e+298) (/ (* 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 t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = x * (y / a);
	} else if (t_1 <= 1e+298) {
		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) :: t_1
    real(8) :: tmp
    t_1 = ((x * y) - (z * t)) / a
    if (t_1 <= (-2d+169)) then
        tmp = x * (y / a)
    else if (t_1 <= 1d+298) 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 t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -2e+169) {
		tmp = x * (y / a);
	} else if (t_1 <= 1e+298) {
		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):
	t_1 = ((x * y) - (z * t)) / a
	tmp = 0
	if t_1 <= -2e+169:
		tmp = x * (y / a)
	elif t_1 <= 1e+298:
		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)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= -2e+169)
		tmp = Float64(x * Float64(y / a));
	elseif (t_1 <= 1e+298)
		tmp = Float64(Float64(x * 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)
	t_1 = ((x * y) - (z * t)) / a;
	tmp = 0.0;
	if (t_1 <= -2e+169)
		tmp = x * (y / a);
	elseif (t_1 <= 1e+298)
		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_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+169], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+298], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -1.99999999999999987e169

   1. Initial program 80.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 49.3

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

   5. Simplified 49.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 50.8

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

   7. Applied egg-rr 50.8%

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

   if -1.99999999999999987e169 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 9.9999999999999996e297

   1. Initial program 98.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 54.6

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

   5. Simplified 54.6%

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

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

   1. Initial program 81.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 33.7

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

   5. Simplified 33.7%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 44.9

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

   7. Applied egg-rr 44.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \leq 10^{+298}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* t (* z (/ -1.0 a)))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 1e+263) (/ (fma (- z) t (* x y)) a) t_1))))

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 = t * (z * (-1.0 / a));
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 1e+263) {
		tmp = fma(-z, t, (x * y)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * Float64(-1.0 / a)))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+263)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a);
	else
		tmp = t_1;
	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[(t * N[(z * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+263], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.00000000000000002e263 < (*.f64 z t)

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 63.8

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

   5. Simplified 63.8%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. /-lowering-/.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000002e263

   1. Initial program 95.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 95.5

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

   4. Applied egg-rr 95.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* t (* z (/ -1.0 a)))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 1e+263) (/ (fma y x (* z (- t))) a) t_1))))

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 = t * (z * (-1.0 / a));
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 1e+263) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * Float64(-1.0 / a)))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+263)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = t_1;
	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[(t * N[(z * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+263], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.00000000000000002e263 < (*.f64 z t)

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 63.8

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

   5. Simplified 63.8%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. /-lowering-/.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000002e263

   1. Initial program 95.5%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 95.5

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

   4. Applied egg-rr 95.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* t (* z (/ -1.0 a)))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 1e+263) (/ (- (* x y) (* z t)) a) t_1))))

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 = t * (z * (-1.0 / a));
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 1e+263) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z * (-1.0 / a));
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((z * t) <= 1e+263) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (z * (-1.0 / a))
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t_1
	elif (z * t) <= 1e+263:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z * Float64(-1.0 / a)))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+263)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z * (-1.0 / a));
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t_1;
	elseif ((z * t) <= 1e+263)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+263], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.00000000000000002e263 < (*.f64 z t)

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 63.8

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

   5. Simplified 63.8%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. /-lowering-/.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000002e263

   1. Initial program 95.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+263}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a}, \frac{x \cdot y}{a}\right)\\ \end{array} \end{array} \]

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 3.7e-63)
   (/ (fma y x (* z (- t))) a)
   (fma (- t) (/ z a) (/ (* x y) a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 3.7e-63) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = fma(-t, (z / a), ((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 (a <= 3.7e-63)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(Float64(-t), Float64(z / a), Float64(Float64(x * y) / 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, 3.7e-63], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) * N[(z / a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.70000000000000012e-63

   1. Initial program 92.3%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.8

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

   4. Applied egg-rr 92.8%

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

   if 3.70000000000000012e-63 < a

   1. Initial program 84.8%

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-neg-in N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 84.4

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

   4. Applied egg-rr 84.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a}, \frac{x \cdot y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;-t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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 (<= (* x y) -5e-29)
   (* x (/ y a))
   (if (<= (* x y) 1e-20) (- (* t (/ z 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 ((x * y) <= -5e-29) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-20) {
		tmp = -(t * (z / 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 ((x * y) <= (-5d-29)) then
        tmp = x * (y / a)
    else if ((x * y) <= 1d-20) then
        tmp = -(t * (z / 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 ((x * y) <= -5e-29) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-20) {
		tmp = -(t * (z / 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 (x * y) <= -5e-29:
		tmp = x * (y / a)
	elif (x * y) <= 1e-20:
		tmp = -(t * (z / 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 (Float64(x * y) <= -5e-29)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= 1e-20)
		tmp = Float64(-Float64(t * Float64(z / 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 ((x * y) <= -5e-29)
		tmp = x * (y / a);
	elseif ((x * y) <= 1e-20)
		tmp = -(t * (z / 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[N[(x * y), $MachinePrecision], -5e-29], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-20], (-N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999986e-29

   1. Initial program 91.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 79.7

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

   7. Applied egg-rr 79.7%

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

   if -4.99999999999999986e-29 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 78.0

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

   5. Simplified 78.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. neg-lowering-neg.f64 80.9

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

   7. Applied egg-rr 80.9%

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

   if 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 79.2

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

   7. Applied egg-rr 79.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;-t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;-z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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 (<= (* x y) -5e-29)
   (* x (/ y a))
   (if (<= (* x y) 1e-20) (- (* z (/ t 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 ((x * y) <= -5e-29) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-20) {
		tmp = -(z * (t / 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 ((x * y) <= (-5d-29)) then
        tmp = x * (y / a)
    else if ((x * y) <= 1d-20) then
        tmp = -(z * (t / 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 ((x * y) <= -5e-29) {
		tmp = x * (y / a);
	} else if ((x * y) <= 1e-20) {
		tmp = -(z * (t / 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 (x * y) <= -5e-29:
		tmp = x * (y / a)
	elif (x * y) <= 1e-20:
		tmp = -(z * (t / 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 (Float64(x * y) <= -5e-29)
		tmp = Float64(x * Float64(y / a));
	elseif (Float64(x * y) <= 1e-20)
		tmp = Float64(-Float64(z * Float64(t / 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 ((x * y) <= -5e-29)
		tmp = x * (y / a);
	elseif ((x * y) <= 1e-20)
		tmp = -(z * (t / 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[N[(x * y), $MachinePrecision], -5e-29], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-20], (-N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999986e-29

   1. Initial program 91.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 79.7

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

   7. Applied egg-rr 79.7%

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

   if -4.99999999999999986e-29 < (*.f64 x y) < 9.99999999999999945e-21

   1. Initial program 87.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 78.0

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

   5. Simplified 78.0%

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

      1. frac-2neg N/A

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

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

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

      3. remove-double-neg N/A

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

      4. associate-*l/ N/A

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

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

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

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

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

      7. neg-lowering-neg.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if 9.99999999999999945e-21 < (*.f64 x y)

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 79.2

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

   7. Applied egg-rr 79.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-20}:\\ \;\;\;\;-z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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 (<= (- (* x y) (* z t)) -1e-24) (* 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 (((x * y) - (z * t)) <= -1e-24) {
		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 (((x * y) - (z * t)) <= (-1d-24)) 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 (((x * y) - (z * t)) <= -1e-24) {
		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 ((x * y) - (z * t)) <= -1e-24:
		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 (Float64(Float64(x * y) - Float64(z * t)) <= -1e-24)
		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 (((x * y) - (z * t)) <= -1e-24)
		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[N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], -1e-24], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.99999999999999924e-25

   1. Initial program 88.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 41.3

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

   5. Simplified 41.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 44.2

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

   7. Applied egg-rr 44.2%

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

   if -9.99999999999999924e-25 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 91.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. *-lowering-*.f64 54.3

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

   5. Simplified 54.3%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 56.9

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

   7. Applied egg-rr 56.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% 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 90.2%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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. *-lowering-*.f64 49.2

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

5. Simplified 49.2%

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

   1. associate-*l/ N/A

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

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

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

   3. /-lowering-/.f64 49.7

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

7. Applied egg-rr 49.7%

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

8. Final simplification 49.7%

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

Developer Target 1: 91.5% 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 2024205 
(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))