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

Percentage Accurate: 92.1% → 97.6%

Time: 9.6s

Alternatives: 11

Speedup: 0.7×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.1% 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: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{a}, y, z \cdot \frac{-t}{a}\right)\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (fma (/ x a) y (* z (/ (- t) a)))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+280) (/ t_2 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 = fma((x / a), y, (z * (-t / a)));
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+280) {
		tmp = t_2 / 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 = fma(Float64(x / a), y, Float64(z * Float64(Float64(-t) / a)))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+280)
		tmp = Float64(t_2 / 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[(N[(x / a), $MachinePrecision] * y + N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+280], N[(t$95$2 / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 5.0000000000000002e280 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 84.1

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

   4. Applied egg-rr 84.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-frac-neg2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 94.4

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

   6. Applied egg-rr 94.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e280

   1. Initial program 98.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \frac{-t}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \frac{-t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \left(t \cdot \frac{-1}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1}{\mathsf{fma}\left(x, -y, z \cdot t\right)}}\\ \end{array} \end{array} \]

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 (<= (- (* x y) (* z t)) -1e+308)
   (fma (/ x a) y (* z (* t (/ -1.0 a))))
   (/ 1.0 (* a (/ -1.0 (fma x (- y) (* z t)))))))

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+308) {
		tmp = fma((x / a), y, (z * (t * (-1.0 / a))));
	} else {
		tmp = 1.0 / (a * (-1.0 / fma(x, -y, (z * t))));
	}
	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+308)
		tmp = fma(Float64(x / a), y, Float64(z * Float64(t * Float64(-1.0 / a))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 / fma(x, Float64(-y), Float64(z * t)))));
	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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], -1e+308], N[(N[(x / a), $MachinePrecision] * y + N[(z * N[(t * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 / N[(x * (-y) + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1e308

   1. Initial program 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 80.1

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

   4. Applied egg-rr 80.1%

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

      1. lift-*.f64 N/A

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

      2. distribute-neg-frac N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-neg.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

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

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

      13. lower-neg.f64 N/A

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

      14. lower-*.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   if -1e308 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 95.5

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

   4. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 95.9

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

   6. Applied egg-rr 95.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \left(t \cdot \frac{-1}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1}{\mathsf{fma}\left(x, -y, z \cdot t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \frac{-t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1}{\mathsf{fma}\left(x, -y, z \cdot t\right)}}\\ \end{array} \end{array} \]

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 (<= (- (* x y) (* z t)) -1e+308)
   (fma (/ x a) y (* z (/ (- t) a)))
   (/ 1.0 (* a (/ -1.0 (fma x (- y) (* z t)))))))

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+308) {
		tmp = fma((x / a), y, (z * (-t / a)));
	} else {
		tmp = 1.0 / (a * (-1.0 / fma(x, -y, (z * t))));
	}
	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+308)
		tmp = fma(Float64(x / a), y, Float64(z * Float64(Float64(-t) / a)));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 / fma(x, Float64(-y), Float64(z * t)))));
	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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], -1e+308], N[(N[(x / a), $MachinePrecision] * y + N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 / N[(x * (-y) + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1e308

   1. Initial program 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 80.1

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

   4. Applied egg-rr 80.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. distribute-frac-neg2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   if -1e308 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 95.5

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

   4. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. distribute-neg-in N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 95.9

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

   6. Applied egg-rr 95.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, z \cdot \frac{-t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1}{\mathsf{fma}\left(x, -y, z \cdot t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+189}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, -\frac{z \cdot t}{a}\right)\\ \end{array} \end{array} \]

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 5.5e+189)
   (/ (- (* x y) (* z t)) a)
   (fma (/ x a) y (- (/ (* z 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 <= 5.5e+189) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = fma((x / a), y, -((z * 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 <= 5.5e+189)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = fma(Float64(x / a), y, Float64(-Float64(Float64(z * 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, 5.5e+189], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + (-N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.5e189

   1. Initial program 95.3%

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

   if 5.5e189 < a

   1. Initial program 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 86.4

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

   4. Applied egg-rr 86.4%

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

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := -\frac{z \cdot t}{a}\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- (/ (* z t) a))))
   (if (<= (* z t) -4e-5) t_1 (if (<= (* z t) 5e-112) (/ (* 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 = -((z * t) / a);
	double tmp;
	if ((z * t) <= -4e-5) {
		tmp = t_1;
	} else if ((z * t) <= 5e-112) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	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 = -((z * t) / a)
    if ((z * t) <= (-4d-5)) then
        tmp = t_1
    else if ((z * t) <= 5d-112) then
        tmp = (x * y) / a
    else
        tmp = t_1
    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 = -((z * t) / a);
	double tmp;
	if ((z * t) <= -4e-5) {
		tmp = t_1;
	} else if ((z * t) <= 5e-112) {
		tmp = (x * y) / 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 = -((z * t) / a)
	tmp = 0
	if (z * t) <= -4e-5:
		tmp = t_1
	elif (z * t) <= 5e-112:
		tmp = (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(-Float64(Float64(z * t) / a))
	tmp = 0.0
	if (Float64(z * t) <= -4e-5)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-112)
		tmp = Float64(Float64(x * y) / 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 = -((z * t) / a);
	tmp = 0.0;
	if ((z * t) <= -4e-5)
		tmp = t_1;
	elseif ((z * t) <= 5e-112)
		tmp = (x * y) / 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[(N[(z * t), $MachinePrecision] / a), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -4e-5], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-112], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.00000000000000033e-5 or 5.00000000000000044e-112 < (*.f64 z t)

   1. Initial program 90.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 76.5

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

   5. Simplified 76.5%

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

   if -4.00000000000000033e-5 < (*.f64 z t) < 5.00000000000000044e-112

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := -t \cdot \frac{z}{a}\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 a)))))
   (if (<= (* z t) -4e-5) t_1 (if (<= (* z t) 5e-112) (/ (* 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 / a));
	double tmp;
	if ((z * t) <= -4e-5) {
		tmp = t_1;
	} else if ((z * t) <= 5e-112) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	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 = -(t * (z / a))
    if ((z * t) <= (-4d-5)) then
        tmp = t_1
    else if ((z * t) <= 5d-112) then
        tmp = (x * y) / a
    else
        tmp = t_1
    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 = -(t * (z / a));
	double tmp;
	if ((z * t) <= -4e-5) {
		tmp = t_1;
	} else if ((z * t) <= 5e-112) {
		tmp = (x * y) / 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 / a))
	tmp = 0
	if (z * t) <= -4e-5:
		tmp = t_1
	elif (z * t) <= 5e-112:
		tmp = (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(-Float64(t * Float64(z / a)))
	tmp = 0.0
	if (Float64(z * t) <= -4e-5)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-112)
		tmp = Float64(Float64(x * y) / 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 / a));
	tmp = 0.0;
	if ((z * t) <= -4e-5)
		tmp = t_1;
	elseif ((z * t) <= 5e-112)
		tmp = (x * y) / 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 / a), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -4e-5], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-112], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.00000000000000033e-5 or 5.00000000000000044e-112 < (*.f64 z t)

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.8

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 76.5

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

   7. Simplified 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 73.9

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

   9. Applied egg-rr 73.9%

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

   if -4.00000000000000033e-5 < (*.f64 z t) < 5.00000000000000044e-112

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;-t \cdot \frac{z}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* z (/ (- t) a))))
   (if (<= (* z t) -2e+37) t_1 (if (<= (* z t) 2e-48) (/ (* 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 = z * (-t / a);
	double tmp;
	if ((z * t) <= -2e+37) {
		tmp = t_1;
	} else if ((z * t) <= 2e-48) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	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 = z * (-t / a)
    if ((z * t) <= (-2d+37)) then
        tmp = t_1
    else if ((z * t) <= 2d-48) then
        tmp = (x * y) / a
    else
        tmp = t_1
    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 = z * (-t / a);
	double tmp;
	if ((z * t) <= -2e+37) {
		tmp = t_1;
	} else if ((z * t) <= 2e-48) {
		tmp = (x * y) / 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 = z * (-t / a)
	tmp = 0
	if (z * t) <= -2e+37:
		tmp = t_1
	elif (z * t) <= 2e-48:
		tmp = (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(z * Float64(Float64(-t) / a))
	tmp = 0.0
	if (Float64(z * t) <= -2e+37)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-48)
		tmp = Float64(Float64(x * y) / 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 = z * (-t / a);
	tmp = 0.0;
	if ((z * t) <= -2e+37)
		tmp = t_1;
	elseif ((z * t) <= 2e-48)
		tmp = (x * y) / 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[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+37], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-48], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999991e37 or 1.9999999999999999e-48 < (*.f64 z t)

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.9

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

   4. Applied egg-rr 90.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 79.3

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

   7. Simplified 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 74.3

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

   9. Applied egg-rr 74.3%

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

   if -1.99999999999999991e37 < (*.f64 z t) < 1.9999999999999999e-48

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 80.0

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

   5. Simplified 80.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \end{array} \end{array} \]

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 (<= (* z t) 5e+290) (/ (- (* x y) (* z t)) a) (* t (* z (/ -1.0 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 ((z * t) <= 5e+290) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t * (z * (-1.0 / 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 ((z * t) <= 5d+290) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t * (z * ((-1.0d0) / 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 ((z * t) <= 5e+290) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t * (z * (-1.0 / 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 (z * t) <= 5e+290:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t * (z * (-1.0 / 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(z * t) <= 5e+290)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(t * Float64(z * Float64(-1.0 / 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 ((z * t) <= 5e+290)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t * (z * (-1.0 / 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[(z * t), $MachinePrecision], 5e+290], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(t * N[(z * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.9999999999999998e290

   1. Initial program 95.0%

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

   if 4.9999999999999998e290 < (*.f64 z t)

   1. Initial program 58.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 58.9

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.0

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

   7. Simplified 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 93.6

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

   9. Applied egg-rr 93.6%

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

       1. neg-mul-1 N/A

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

       2. associate-*l/ N/A

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

       3. lift-/.f64 N/A

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

       4. lower-*.f64 93.8

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

   11. Applied egg-rr 93.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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
 (if (<= t 1.3e-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 (t <= 1.3e-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 (t <= 1.3d-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 (t <= 1.3e-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 t <= 1.3e-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 (t <= 1.3e-24)
		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)
	tmp = 0.0;
	if (t <= 1.3e-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[t, 1.3e-24], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-24

   1. Initial program 92.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 60.0

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

   5. Simplified 60.0%

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

   if 1.3e-24 < t

   1. Initial program 93.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.2

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

   5. Simplified 35.2%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 39.9

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

   7. Applied egg-rr 39.9%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

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 (<= z -2.5e-280) (* 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 (z <= -2.5e-280) {
		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 (z <= (-2.5d-280)) 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 (z <= -2.5e-280) {
		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 z <= -2.5e-280:
		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 (z <= -2.5e-280)
		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 (z <= -2.5e-280)
		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[z, -2.5e-280], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000014e-280

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 52.1

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

   5. Simplified 52.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 50.5

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

   7. Applied egg-rr 50.5%

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

   if -2.50000000000000014e-280 < z

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 53.3

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

   5. Simplified 53.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 55.6

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

   7. Applied egg-rr 55.6%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.1% 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 93.0%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 52.7

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

5. Simplified 52.7%

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

   1. associate-*l/ N/A

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

   2. lift-/.f64 N/A

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

   3. lower-*.f64 53.5

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

7. Applied egg-rr 53.5%

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

8. Final simplification 53.5%

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

Developer Target 1: 91.4% 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 2024219 
(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))