Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Percentage Accurate: 91.3% → 92.9%

Time: 9.1s

Alternatives: 12

Speedup: 0.6×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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 * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := z \cdot \left(t \cdot -9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_1\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{0.5}{a\_m} + x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* z (* t -9.0))))
   (*
    a_s
    (if (<= (* a_m 2.0) 1e+66)
      (/ (fma x y t_1) (* a_m 2.0))
      (+ (* t_1 (/ 0.5 a_m)) (* x (* y (/ 0.5 a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z * (t * -9.0);
	double tmp;
	if ((a_m * 2.0) <= 1e+66) {
		tmp = fma(x, y, t_1) / (a_m * 2.0);
	} else {
		tmp = (t_1 * (0.5 / a_m)) + (x * (y * (0.5 / a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(z * Float64(t * -9.0))
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 1e+66)
		tmp = Float64(fma(x, y, t_1) / Float64(a_m * 2.0));
	else
		tmp = Float64(Float64(t_1 * Float64(0.5 / a_m)) + Float64(x * Float64(y * Float64(0.5 / a_m))));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 1e+66], N[(N[(x * y + t$95$1), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 9.99999999999999945e65

   1. Initial program 94.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. div-sub 91.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      2. *-commutative 91.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 94.5%

         \[\leadsto \color{blue}{\frac{y \cdot x - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. cancel-sign-sub-inv 94.5%

         \[\leadsto \frac{\color{blue}{y \cdot x + \left(-z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. *-commutative 94.5%

         \[\leadsto \frac{\color{blue}{x \cdot y} + \left(-z \cdot 9\right) \cdot t}{a \cdot 2} \]

      6. fma-define 94.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, \left(-z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]

      7. distribute-rgt-neg-in 94.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{\left(z \cdot \left(-9\right)\right)} \cdot t\right)}{a \cdot 2} \]

      8. associate-*r* 94.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(\left(-9\right) \cdot t\right)}\right)}{a \cdot 2} \]

      9. distribute-lft-neg-in 94.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(-9 \cdot t\right)}\right)}{a \cdot 2} \]

      10. *-commutative 94.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(-\color{blue}{t \cdot 9}\right)\right)}{a \cdot 2} \]

      11. distribute-rgt-neg-in 94.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot \left(-9\right)\right)}\right)}{a \cdot 2} \]

      12. metadata-eval 94.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot \color{blue}{-9}\right)\right)}{a \cdot 2} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}} \]
   4. Add Preprocessing

   if 9.99999999999999945e65 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 79.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 78.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 78.2%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 78.2%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 78.2%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 78.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 78.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 78.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 78.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 78.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 78.2%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 78.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 78.2%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 78.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 78.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 78.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 78.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 79.3%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)} \]

      2. fma-undefine 79.3%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

      3. distribute-lft-in 79.3%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right) + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)} \]

      4. *-commutative 79.3%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a}} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right) \]

      5. associate-*l* 91.5%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right) \]

   8. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a} + x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot -9\right)}{a\_m \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a\_m} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a\_m}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -5e+145)
    (* x (* y (/ 0.5 a_m)))
    (if (<= (* x y) -2e+57)
      (/ (* z (* t -9.0)) (* a_m 2.0))
      (if (<= (* x y) -2e-32)
        (* (/ 0.5 a_m) (* x y))
        (if (<= (* x y) 5e-30)
          (/ (* z -4.5) (/ a_m t))
          (* x (/ (* y 0.5) a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z * (t * -9.0)) / (a_m * 2.0);
	} else if ((x * y) <= -2e-32) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = (z * -4.5) / (a_m / t);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-5d+145)) then
        tmp = x * (y * (0.5d0 / a_m))
    else if ((x * y) <= (-2d+57)) then
        tmp = (z * (t * (-9.0d0))) / (a_m * 2.0d0)
    else if ((x * y) <= (-2d-32)) then
        tmp = (0.5d0 / a_m) * (x * y)
    else if ((x * y) <= 5d-30) then
        tmp = (z * (-4.5d0)) / (a_m / t)
    else
        tmp = x * ((y * 0.5d0) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z * (t * -9.0)) / (a_m * 2.0);
	} else if ((x * y) <= -2e-32) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = (z * -4.5) / (a_m / t);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -5e+145:
		tmp = x * (y * (0.5 / a_m))
	elif (x * y) <= -2e+57:
		tmp = (z * (t * -9.0)) / (a_m * 2.0)
	elif (x * y) <= -2e-32:
		tmp = (0.5 / a_m) * (x * y)
	elif (x * y) <= 5e-30:
		tmp = (z * -4.5) / (a_m / t)
	else:
		tmp = x * ((y * 0.5) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -5e+145)
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	elseif (Float64(x * y) <= -2e+57)
		tmp = Float64(Float64(z * Float64(t * -9.0)) / Float64(a_m * 2.0));
	elseif (Float64(x * y) <= -2e-32)
		tmp = Float64(Float64(0.5 / a_m) * Float64(x * y));
	elseif (Float64(x * y) <= 5e-30)
		tmp = Float64(Float64(z * -4.5) / Float64(a_m / t));
	else
		tmp = Float64(x * Float64(Float64(y * 0.5) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -5e+145)
		tmp = x * (y * (0.5 / a_m));
	elseif ((x * y) <= -2e+57)
		tmp = (z * (t * -9.0)) / (a_m * 2.0);
	elseif ((x * y) <= -2e-32)
		tmp = (0.5 / a_m) * (x * y);
	elseif ((x * y) <= 5e-30)
		tmp = (z * -4.5) / (a_m / t);
	else
		tmp = x * ((y * 0.5) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+145], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+57], N[(N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-32], N[(N[(0.5 / a$95$m), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-30], N[(N[(z * -4.5), $MachinePrecision] / N[(a$95$m / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -4.99999999999999967e145

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 88.6%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 88.6%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 88.6%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 88.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 88.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 72.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 72.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 72.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 83.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 83.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 83.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 83.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   if -4.99999999999999967e145 < (*.f64 x y) < -2.0000000000000001e57

   1. Initial program 99.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.6%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right)} \cdot -9}{a \cdot 2} \]

      3. associate-*r* 77.7%

         \[\leadsto \frac{\color{blue}{z \cdot \left(t \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{z \cdot \left(t \cdot -9\right)}}{a \cdot 2} \]

   if -2.0000000000000001e57 < (*.f64 x y) < -2.00000000000000011e-32

   1. Initial program 99.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      2. fma-neg 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      3. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      4. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      5. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      6. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      7. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      8. associate-/r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if -2.00000000000000011e-32 < (*.f64 x y) < 4.99999999999999972e-30

   1. Initial program 94.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. div-inv 83.6%

         \[\leadsto -4.5 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{a}\right)} \]

      2. *-commutative 83.6%

         \[\leadsto -4.5 \cdot \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{a}\right) \]

      3. associate-*l* 79.2%

         \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \left(t \cdot \frac{1}{a}\right)\right)} \]

      4. div-inv 79.2%

         \[\leadsto -4.5 \cdot \left(z \cdot \color{blue}{\frac{t}{a}}\right) \]

      5. associate-*l* 79.2%

         \[\leadsto \color{blue}{\left(-4.5 \cdot z\right) \cdot \frac{t}{a}} \]

      6. clear-num 79.3%

         \[\leadsto \left(-4.5 \cdot z\right) \cdot \color{blue}{\frac{1}{\frac{a}{t}}} \]

      7. un-div-inv 79.9%

         \[\leadsto \color{blue}{\frac{-4.5 \cdot z}{\frac{a}{t}}} \]

      8. *-commutative 79.9%

         \[\leadsto \frac{\color{blue}{z \cdot -4.5}}{\frac{a}{t}} \]

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{z \cdot -4.5}{\frac{a}{t}}} \]

   if 4.99999999999999972e-30 < (*.f64 x y)

   1. Initial program 84.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 76.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 76.3%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 76.3%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 76.3%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a\_m} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a\_m}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -5e+145)
    (* x (* y (/ 0.5 a_m)))
    (if (<= (* x y) -2e+57)
      (* (/ z a_m) (* t -4.5))
      (if (<= (* x y) -2e-32)
        (* (/ 0.5 a_m) (* x y))
        (if (<= (* x y) 5e-30)
          (/ (* z -4.5) (/ a_m t))
          (* x (/ (* y 0.5) a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-32) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = (z * -4.5) / (a_m / t);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-5d+145)) then
        tmp = x * (y * (0.5d0 / a_m))
    else if ((x * y) <= (-2d+57)) then
        tmp = (z / a_m) * (t * (-4.5d0))
    else if ((x * y) <= (-2d-32)) then
        tmp = (0.5d0 / a_m) * (x * y)
    else if ((x * y) <= 5d-30) then
        tmp = (z * (-4.5d0)) / (a_m / t)
    else
        tmp = x * ((y * 0.5d0) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-32) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = (z * -4.5) / (a_m / t);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -5e+145:
		tmp = x * (y * (0.5 / a_m))
	elif (x * y) <= -2e+57:
		tmp = (z / a_m) * (t * -4.5)
	elif (x * y) <= -2e-32:
		tmp = (0.5 / a_m) * (x * y)
	elif (x * y) <= 5e-30:
		tmp = (z * -4.5) / (a_m / t)
	else:
		tmp = x * ((y * 0.5) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -5e+145)
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	elseif (Float64(x * y) <= -2e+57)
		tmp = Float64(Float64(z / a_m) * Float64(t * -4.5));
	elseif (Float64(x * y) <= -2e-32)
		tmp = Float64(Float64(0.5 / a_m) * Float64(x * y));
	elseif (Float64(x * y) <= 5e-30)
		tmp = Float64(Float64(z * -4.5) / Float64(a_m / t));
	else
		tmp = Float64(x * Float64(Float64(y * 0.5) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -5e+145)
		tmp = x * (y * (0.5 / a_m));
	elseif ((x * y) <= -2e+57)
		tmp = (z / a_m) * (t * -4.5);
	elseif ((x * y) <= -2e-32)
		tmp = (0.5 / a_m) * (x * y);
	elseif ((x * y) <= 5e-30)
		tmp = (z * -4.5) / (a_m / t);
	else
		tmp = x * ((y * 0.5) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+145], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+57], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-32], N[(N[(0.5 / a$95$m), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-30], N[(N[(z * -4.5), $MachinePrecision] / N[(a$95$m / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -4.99999999999999967e145

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 88.6%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 88.6%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 88.6%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 88.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 88.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 72.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 72.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 72.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 83.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 83.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 83.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 83.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   if -4.99999999999999967e145 < (*.f64 x y) < -2.0000000000000001e57

   1. Initial program 99.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.6%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 77.8%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 77.8%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. metadata-eval 77.4%

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{-4.5} \]

      4. *-commutative 77.4%

         \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

      5. associate-/l* 77.4%

         \[\leadsto -4.5 \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      6. associate-*r* 77.7%

         \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   if -2.0000000000000001e57 < (*.f64 x y) < -2.00000000000000011e-32

   1. Initial program 99.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      2. fma-neg 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      3. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      4. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      5. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      6. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      7. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      8. associate-/r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      9. metadata-eval 99.7%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if -2.00000000000000011e-32 < (*.f64 x y) < 4.99999999999999972e-30

   1. Initial program 94.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.6%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. div-inv 83.6%

         \[\leadsto -4.5 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot \frac{1}{a}\right)} \]

      2. *-commutative 83.6%

         \[\leadsto -4.5 \cdot \left(\color{blue}{\left(z \cdot t\right)} \cdot \frac{1}{a}\right) \]

      3. associate-*l* 79.2%

         \[\leadsto -4.5 \cdot \color{blue}{\left(z \cdot \left(t \cdot \frac{1}{a}\right)\right)} \]

      4. div-inv 79.2%

         \[\leadsto -4.5 \cdot \left(z \cdot \color{blue}{\frac{t}{a}}\right) \]

      5. associate-*l* 79.2%

         \[\leadsto \color{blue}{\left(-4.5 \cdot z\right) \cdot \frac{t}{a}} \]

      6. clear-num 79.3%

         \[\leadsto \left(-4.5 \cdot z\right) \cdot \color{blue}{\frac{1}{\frac{a}{t}}} \]

      7. un-div-inv 79.9%

         \[\leadsto \color{blue}{\frac{-4.5 \cdot z}{\frac{a}{t}}} \]

      8. *-commutative 79.9%

         \[\leadsto \frac{\color{blue}{z \cdot -4.5}}{\frac{a}{t}} \]

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{z \cdot -4.5}{\frac{a}{t}}} \]

   if 4.99999999999999972e-30 < (*.f64 x y)

   1. Initial program 84.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 76.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 76.3%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 76.3%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 76.3%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5}{\frac{a\_m}{x \cdot y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -5e+145)
    (* x (* y (/ 0.5 a_m)))
    (if (<= (* x y) -2e+57)
      (* (/ z a_m) (* t -4.5))
      (if (<= (* x y) -2e-9)
        (/ 0.5 (/ a_m (* x y)))
        (if (<= (* x y) 5e-30)
          (* -4.5 (/ (* z t) a_m))
          (* x (/ (* y 0.5) a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-9) {
		tmp = 0.5 / (a_m / (x * y));
	} else if ((x * y) <= 5e-30) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-5d+145)) then
        tmp = x * (y * (0.5d0 / a_m))
    else if ((x * y) <= (-2d+57)) then
        tmp = (z / a_m) * (t * (-4.5d0))
    else if ((x * y) <= (-2d-9)) then
        tmp = 0.5d0 / (a_m / (x * y))
    else if ((x * y) <= 5d-30) then
        tmp = (-4.5d0) * ((z * t) / a_m)
    else
        tmp = x * ((y * 0.5d0) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-9) {
		tmp = 0.5 / (a_m / (x * y));
	} else if ((x * y) <= 5e-30) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -5e+145:
		tmp = x * (y * (0.5 / a_m))
	elif (x * y) <= -2e+57:
		tmp = (z / a_m) * (t * -4.5)
	elif (x * y) <= -2e-9:
		tmp = 0.5 / (a_m / (x * y))
	elif (x * y) <= 5e-30:
		tmp = -4.5 * ((z * t) / a_m)
	else:
		tmp = x * ((y * 0.5) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -5e+145)
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	elseif (Float64(x * y) <= -2e+57)
		tmp = Float64(Float64(z / a_m) * Float64(t * -4.5));
	elseif (Float64(x * y) <= -2e-9)
		tmp = Float64(0.5 / Float64(a_m / Float64(x * y)));
	elseif (Float64(x * y) <= 5e-30)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a_m));
	else
		tmp = Float64(x * Float64(Float64(y * 0.5) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -5e+145)
		tmp = x * (y * (0.5 / a_m));
	elseif ((x * y) <= -2e+57)
		tmp = (z / a_m) * (t * -4.5);
	elseif ((x * y) <= -2e-9)
		tmp = 0.5 / (a_m / (x * y));
	elseif ((x * y) <= 5e-30)
		tmp = -4.5 * ((z * t) / a_m);
	else
		tmp = x * ((y * 0.5) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+145], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+57], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-9], N[(0.5 / N[(a$95$m / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-30], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -4.99999999999999967e145

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 88.6%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 88.6%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 88.6%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 88.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 88.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 72.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 72.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 72.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 83.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 83.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 83.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 83.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   if -4.99999999999999967e145 < (*.f64 x y) < -2.0000000000000001e57

   1. Initial program 99.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.6%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 77.8%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 77.8%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. metadata-eval 77.4%

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{-4.5} \]

      4. *-commutative 77.4%

         \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

      5. associate-/l* 77.4%

         \[\leadsto -4.5 \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      6. associate-*r* 77.7%

         \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   if -2.0000000000000001e57 < (*.f64 x y) < -2.00000000000000012e-9

   1. Initial program 99.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 99.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 99.5%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 99.5%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 99.5%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 99.5%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 99.5%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 99.5%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 99.5%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 99.5%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 99.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 99.5%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 99.5%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 99.5%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 99.5%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 94.1%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{x \cdot y}}} \]

   if -2.00000000000000012e-9 < (*.f64 x y) < 4.99999999999999972e-30

   1. Initial program 94.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 4.99999999999999972e-30 < (*.f64 x y)

   1. Initial program 84.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 76.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 76.3%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 76.3%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 76.3%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5}{\frac{a}{x \cdot y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5}{a\_m} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -5e+145)
    (* x (* y (/ 0.5 a_m)))
    (if (<= (* x y) -2e+57)
      (* (/ z a_m) (* t -4.5))
      (if (<= (* x y) -2e-9)
        (* (/ 0.5 a_m) (* x y))
        (if (<= (* x y) 5e-30)
          (* -4.5 (/ (* z t) a_m))
          (* x (/ (* y 0.5) a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-9) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-5d+145)) then
        tmp = x * (y * (0.5d0 / a_m))
    else if ((x * y) <= (-2d+57)) then
        tmp = (z / a_m) * (t * (-4.5d0))
    else if ((x * y) <= (-2d-9)) then
        tmp = (0.5d0 / a_m) * (x * y)
    else if ((x * y) <= 5d-30) then
        tmp = (-4.5d0) * ((z * t) / a_m)
    else
        tmp = x * ((y * 0.5d0) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+145) {
		tmp = x * (y * (0.5 / a_m));
	} else if ((x * y) <= -2e+57) {
		tmp = (z / a_m) * (t * -4.5);
	} else if ((x * y) <= -2e-9) {
		tmp = (0.5 / a_m) * (x * y);
	} else if ((x * y) <= 5e-30) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * ((y * 0.5) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -5e+145:
		tmp = x * (y * (0.5 / a_m))
	elif (x * y) <= -2e+57:
		tmp = (z / a_m) * (t * -4.5)
	elif (x * y) <= -2e-9:
		tmp = (0.5 / a_m) * (x * y)
	elif (x * y) <= 5e-30:
		tmp = -4.5 * ((z * t) / a_m)
	else:
		tmp = x * ((y * 0.5) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -5e+145)
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	elseif (Float64(x * y) <= -2e+57)
		tmp = Float64(Float64(z / a_m) * Float64(t * -4.5));
	elseif (Float64(x * y) <= -2e-9)
		tmp = Float64(Float64(0.5 / a_m) * Float64(x * y));
	elseif (Float64(x * y) <= 5e-30)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a_m));
	else
		tmp = Float64(x * Float64(Float64(y * 0.5) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -5e+145)
		tmp = x * (y * (0.5 / a_m));
	elseif ((x * y) <= -2e+57)
		tmp = (z / a_m) * (t * -4.5);
	elseif ((x * y) <= -2e-9)
		tmp = (0.5 / a_m) * (x * y);
	elseif ((x * y) <= 5e-30)
		tmp = -4.5 * ((z * t) / a_m);
	else
		tmp = x * ((y * 0.5) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+145], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+57], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-9], N[(N[(0.5 / a$95$m), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-30], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -4.99999999999999967e145

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 88.6%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 88.6%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 88.6%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 88.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 88.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 88.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 88.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 88.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 72.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 72.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 72.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 83.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 83.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 83.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 83.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   if -4.99999999999999967e145 < (*.f64 x y) < -2.0000000000000001e57

   1. Initial program 99.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.6%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 77.8%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 77.8%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      3. metadata-eval 77.4%

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{-4.5} \]

      4. *-commutative 77.4%

         \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

      5. associate-/l* 77.4%

         \[\leadsto -4.5 \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      6. associate-*r* 77.7%

         \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(-4.5 \cdot t\right) \cdot \frac{z}{a}} \]

   if -2.0000000000000001e57 < (*.f64 x y) < -2.00000000000000012e-9

   1. Initial program 99.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.2%

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      2. fma-neg 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      3. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      4. distribute-rgt-neg-in 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      5. distribute-rgt-neg-in 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      6. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      7. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      8. associate-/r* 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      9. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in x around inf 93.9%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if -2.00000000000000012e-9 < (*.f64 x y) < 4.99999999999999972e-30

   1. Initial program 94.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 4.99999999999999972e-30 < (*.f64 x y)

   1. Initial program 84.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 76.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 76.3%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 76.3%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 76.3%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+145}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-30}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+88} \lor \neg \left(x \leq 1.85 \cdot 10^{-119}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* x (* y (/ 0.5 a_m)))))
   (*
    a_s
    (if (<= x -5.6e+134)
      t_1
      (if (<= x -4.5e+118)
        (* t (* -4.5 (/ z a_m)))
        (if (or (<= x -5e+88) (not (<= x 1.85e-119)))
          t_1
          (* -4.5 (/ (* z t) a_m))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * (y * (0.5 / a_m));
	double tmp;
	if (x <= -5.6e+134) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if ((x <= -5e+88) || !(x <= 1.85e-119)) {
		tmp = t_1;
	} else {
		tmp = -4.5 * ((z * t) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * (0.5d0 / a_m))
    if (x <= (-5.6d+134)) then
        tmp = t_1
    else if (x <= (-4.5d+118)) then
        tmp = t * ((-4.5d0) * (z / a_m))
    else if ((x <= (-5d+88)) .or. (.not. (x <= 1.85d-119))) then
        tmp = t_1
    else
        tmp = (-4.5d0) * ((z * t) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * (y * (0.5 / a_m));
	double tmp;
	if (x <= -5.6e+134) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if ((x <= -5e+88) || !(x <= 1.85e-119)) {
		tmp = t_1;
	} else {
		tmp = -4.5 * ((z * t) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = x * (y * (0.5 / a_m))
	tmp = 0
	if x <= -5.6e+134:
		tmp = t_1
	elif x <= -4.5e+118:
		tmp = t * (-4.5 * (z / a_m))
	elif (x <= -5e+88) or not (x <= 1.85e-119):
		tmp = t_1
	else:
		tmp = -4.5 * ((z * t) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(x * Float64(y * Float64(0.5 / a_m)))
	tmp = 0.0
	if (x <= -5.6e+134)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	elseif ((x <= -5e+88) || !(x <= 1.85e-119))
		tmp = t_1;
	else
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = x * (y * (0.5 / a_m));
	tmp = 0.0;
	if (x <= -5.6e+134)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = t * (-4.5 * (z / a_m));
	elseif ((x <= -5e+88) || ~((x <= 1.85e-119)))
		tmp = t_1;
	else
		tmp = -4.5 * ((z * t) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -5.6e+134], t$95$1, If[LessEqual[x, -4.5e+118], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -5e+88], N[Not[LessEqual[x, 1.85e-119]], $MachinePrecision]], t$95$1, N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5999999999999997e134 or -4.50000000000000002e118 < x < -4.99999999999999997e88 or 1.8500000000000001e-119 < x

   1. Initial program 88.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 88.2%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 88.2%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 88.2%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 88.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 88.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 88.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 88.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 88.2%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 88.2%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 88.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 88.2%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 88.2%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 88.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 88.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 88.2%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 61.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 61.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 61.0%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 61.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 69.1%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 69.1%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 69.1%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 69.1%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 69.1%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   if -5.5999999999999997e134 < x < -4.50000000000000002e118

   1. Initial program 53.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 27.1%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 27.1%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 27.5%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 27.5%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \color{blue}{t \cdot \frac{z \cdot -9}{a \cdot 2}} \]

      2. *-commutative 50.6%

         \[\leadsto t \cdot \frac{\color{blue}{-9 \cdot z}}{a \cdot 2} \]

      3. *-commutative 50.6%

         \[\leadsto t \cdot \frac{-9 \cdot z}{\color{blue}{2 \cdot a}} \]

      4. times-frac 50.3%

         \[\leadsto t \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right)} \]

      5. metadata-eval 50.3%

         \[\leadsto t \cdot \left(\color{blue}{-4.5} \cdot \frac{z}{a}\right) \]

   7. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a}\right)} \]

   if -4.99999999999999997e88 < x < 1.8500000000000001e-119

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+88} \lor \neg \left(x \leq 1.85 \cdot 10^{-119}\right):\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a_m))))
   (*
    a_s
    (if (<= x -4.9e+134)
      t_1
      (if (<= x -4.5e+118)
        (* t (* -4.5 (/ z a_m)))
        (if (<= x -7e+91)
          t_1
          (if (<= x 4.6e-121)
            (* z (/ (* t -4.5) a_m))
            (* x (* y (/ 0.5 a_m))))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * ((y * 0.5) / a_m);
	double tmp;
	if (x <= -4.9e+134) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if (x <= -7e+91) {
		tmp = t_1;
	} else if (x <= 4.6e-121) {
		tmp = z * ((t * -4.5) / a_m);
	} else {
		tmp = x * (y * (0.5 / a_m));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a_m)
    if (x <= (-4.9d+134)) then
        tmp = t_1
    else if (x <= (-4.5d+118)) then
        tmp = t * ((-4.5d0) * (z / a_m))
    else if (x <= (-7d+91)) then
        tmp = t_1
    else if (x <= 4.6d-121) then
        tmp = z * ((t * (-4.5d0)) / a_m)
    else
        tmp = x * (y * (0.5d0 / a_m))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * ((y * 0.5) / a_m);
	double tmp;
	if (x <= -4.9e+134) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if (x <= -7e+91) {
		tmp = t_1;
	} else if (x <= 4.6e-121) {
		tmp = z * ((t * -4.5) / a_m);
	} else {
		tmp = x * (y * (0.5 / a_m));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = x * ((y * 0.5) / a_m)
	tmp = 0
	if x <= -4.9e+134:
		tmp = t_1
	elif x <= -4.5e+118:
		tmp = t * (-4.5 * (z / a_m))
	elif x <= -7e+91:
		tmp = t_1
	elif x <= 4.6e-121:
		tmp = z * ((t * -4.5) / a_m)
	else:
		tmp = x * (y * (0.5 / a_m))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a_m))
	tmp = 0.0
	if (x <= -4.9e+134)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	elseif (x <= -7e+91)
		tmp = t_1;
	elseif (x <= 4.6e-121)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a_m));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = x * ((y * 0.5) / a_m);
	tmp = 0.0;
	if (x <= -4.9e+134)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = t * (-4.5 * (z / a_m));
	elseif (x <= -7e+91)
		tmp = t_1;
	elseif (x <= 4.6e-121)
		tmp = z * ((t * -4.5) / a_m);
	else
		tmp = x * (y * (0.5 / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -4.9e+134], t$95$1, If[LessEqual[x, -4.5e+118], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e+91], t$95$1, If[LessEqual[x, 4.6e-121], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -4.89999999999999996e134 or -4.50000000000000002e118 < x < -7.00000000000000001e91

   1. Initial program 82.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 77.2%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 77.2%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 77.2%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 77.2%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]

   if -4.89999999999999996e134 < x < -4.50000000000000002e118

   1. Initial program 53.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 27.1%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 27.1%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 27.5%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 27.5%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \color{blue}{t \cdot \frac{z \cdot -9}{a \cdot 2}} \]

      2. *-commutative 50.6%

         \[\leadsto t \cdot \frac{\color{blue}{-9 \cdot z}}{a \cdot 2} \]

      3. *-commutative 50.6%

         \[\leadsto t \cdot \frac{-9 \cdot z}{\color{blue}{2 \cdot a}} \]

      4. times-frac 50.3%

         \[\leadsto t \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right)} \]

      5. metadata-eval 50.3%

         \[\leadsto t \cdot \left(\color{blue}{-4.5} \cdot \frac{z}{a}\right) \]

   7. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a}\right)} \]

   if -7.00000000000000001e91 < x < 4.60000000000000025e-121

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.9%

         \[\leadsto \color{blue}{\frac{-4.5 \cdot \left(t \cdot z\right)}{a}} \]

      2. associate-*r* 68.0%

         \[\leadsto \frac{\color{blue}{\left(-4.5 \cdot t\right) \cdot z}}{a} \]

      3. associate-*l/ 63.1%

         \[\leadsto \color{blue}{\frac{-4.5 \cdot t}{a} \cdot z} \]

      4. associate-*r/ 63.1%

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{t}{a}\right)} \cdot z \]

      5. *-commutative 63.1%

         \[\leadsto \color{blue}{z \cdot \left(-4.5 \cdot \frac{t}{a}\right)} \]

      6. associate-*r/ 63.1%

         \[\leadsto z \cdot \color{blue}{\frac{-4.5 \cdot t}{a}} \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{z \cdot \frac{-4.5 \cdot t}{a}} \]

   if 4.60000000000000025e-121 < x

   1. Initial program 92.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 91.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 91.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 91.5%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 91.5%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 91.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 91.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 91.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 58.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 58.7%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 58.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 64.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 64.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 64.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 64.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-121}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-119}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a_m))))
   (*
    a_s
    (if (<= x -1.1e+135)
      t_1
      (if (<= x -4.5e+118)
        (* t (* -4.5 (/ z a_m)))
        (if (<= x -5.7e+88)
          t_1
          (if (<= x 1.3e-119)
            (* -4.5 (/ (* z t) a_m))
            (* x (* y (/ 0.5 a_m))))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * ((y * 0.5) / a_m);
	double tmp;
	if (x <= -1.1e+135) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if (x <= -5.7e+88) {
		tmp = t_1;
	} else if (x <= 1.3e-119) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * (y * (0.5 / a_m));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a_m)
    if (x <= (-1.1d+135)) then
        tmp = t_1
    else if (x <= (-4.5d+118)) then
        tmp = t * ((-4.5d0) * (z / a_m))
    else if (x <= (-5.7d+88)) then
        tmp = t_1
    else if (x <= 1.3d-119) then
        tmp = (-4.5d0) * ((z * t) / a_m)
    else
        tmp = x * (y * (0.5d0 / a_m))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = x * ((y * 0.5) / a_m);
	double tmp;
	if (x <= -1.1e+135) {
		tmp = t_1;
	} else if (x <= -4.5e+118) {
		tmp = t * (-4.5 * (z / a_m));
	} else if (x <= -5.7e+88) {
		tmp = t_1;
	} else if (x <= 1.3e-119) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = x * (y * (0.5 / a_m));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = x * ((y * 0.5) / a_m)
	tmp = 0
	if x <= -1.1e+135:
		tmp = t_1
	elif x <= -4.5e+118:
		tmp = t * (-4.5 * (z / a_m))
	elif x <= -5.7e+88:
		tmp = t_1
	elif x <= 1.3e-119:
		tmp = -4.5 * ((z * t) / a_m)
	else:
		tmp = x * (y * (0.5 / a_m))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a_m))
	tmp = 0.0
	if (x <= -1.1e+135)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	elseif (x <= -5.7e+88)
		tmp = t_1;
	elseif (x <= 1.3e-119)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a_m));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = x * ((y * 0.5) / a_m);
	tmp = 0.0;
	if (x <= -1.1e+135)
		tmp = t_1;
	elseif (x <= -4.5e+118)
		tmp = t * (-4.5 * (z / a_m));
	elseif (x <= -5.7e+88)
		tmp = t_1;
	elseif (x <= 1.3e-119)
		tmp = -4.5 * ((z * t) / a_m);
	else
		tmp = x * (y * (0.5 / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -1.1e+135], t$95$1, If[LessEqual[x, -4.5e+118], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.7e+88], t$95$1, If[LessEqual[x, 1.3e-119], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -1.1e135 or -4.50000000000000002e118 < x < -5.70000000000000021e88

   1. Initial program 82.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-/l* 77.2%

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{a}\right)} \cdot 0.5 \]

      3. associate-*r* 77.2%

         \[\leadsto \color{blue}{x \cdot \left(\frac{y}{a} \cdot 0.5\right)} \]

      4. *-commutative 77.2%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{a}\right)} \]

      5. associate-*r/ 77.2%

         \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot y}{a}} \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{x \cdot \frac{0.5 \cdot y}{a}} \]

   if -1.1e135 < x < -4.50000000000000002e118

   1. Initial program 53.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 27.1%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 27.1%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 27.5%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 27.5%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \color{blue}{t \cdot \frac{z \cdot -9}{a \cdot 2}} \]

      2. *-commutative 50.6%

         \[\leadsto t \cdot \frac{\color{blue}{-9 \cdot z}}{a \cdot 2} \]

      3. *-commutative 50.6%

         \[\leadsto t \cdot \frac{-9 \cdot z}{\color{blue}{2 \cdot a}} \]

      4. times-frac 50.3%

         \[\leadsto t \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right)} \]

      5. metadata-eval 50.3%

         \[\leadsto t \cdot \left(\color{blue}{-4.5} \cdot \frac{z}{a}\right) \]

   7. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a}\right)} \]

   if -5.70000000000000021e88 < x < 1.30000000000000006e-119

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 1.30000000000000006e-119 < x

   1. Initial program 92.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 91.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 91.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 91.5%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 91.5%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 91.6%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 91.6%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 91.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 91.6%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 91.6%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 91.6%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   7. Taylor expanded in x around inf 58.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} \cdot 0.5} \]

      2. associate-*l/ 58.7%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{a}} \]

      3. associate-*r* 58.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot 0.5\right)}}{a} \]

      4. associate-/l* 64.4%

         \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot 0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 64.3%

          \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{0.5}{a}\right)} \]

       2. *-commutative 64.3%

          \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]

   11. Applied egg-rr 64.3%

       \[\leadsto x \cdot \color{blue}{\left(\frac{0.5}{a} \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-119}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a\_m} + x \cdot \left(y \cdot \frac{0.5}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* a_m 2.0) 5e-21)
    (/ (- (* x y) (* t (* z 9.0))) (* a_m 2.0))
    (+ (* (* z (* t -9.0)) (/ 0.5 a_m)) (* x (* y (/ 0.5 a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 5e-21) {
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	} else {
		tmp = ((z * (t * -9.0)) * (0.5 / a_m)) + (x * (y * (0.5 / a_m)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((a_m * 2.0d0) <= 5d-21) then
        tmp = ((x * y) - (t * (z * 9.0d0))) / (a_m * 2.0d0)
    else
        tmp = ((z * (t * (-9.0d0))) * (0.5d0 / a_m)) + (x * (y * (0.5d0 / a_m)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((a_m * 2.0) <= 5e-21) {
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	} else {
		tmp = ((z * (t * -9.0)) * (0.5 / a_m)) + (x * (y * (0.5 / a_m)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (a_m * 2.0) <= 5e-21:
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0)
	else:
		tmp = ((z * (t * -9.0)) * (0.5 / a_m)) + (x * (y * (0.5 / a_m)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(a_m * 2.0) <= 5e-21)
		tmp = Float64(Float64(Float64(x * y) - Float64(t * Float64(z * 9.0))) / Float64(a_m * 2.0));
	else
		tmp = Float64(Float64(Float64(z * Float64(t * -9.0)) * Float64(0.5 / a_m)) + Float64(x * Float64(y * Float64(0.5 / a_m))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((a_m * 2.0) <= 5e-21)
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	else
		tmp = ((z * (t * -9.0)) * (0.5 / a_m)) + (x * (y * (0.5 / a_m)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(a$95$m * 2.0), $MachinePrecision], 5e-21], N[(N[(N[(x * y), $MachinePrecision] - N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.99999999999999973e-21

   1. Initial program 94.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   if 4.99999999999999973e-21 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 83.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 82.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}} \]

      2. inv-pow 82.3%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1}} \]

      3. *-commutative 82.3%

         \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}^{-1} \]

      4. associate-/l* 82.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \frac{a}{x \cdot y - \left(z \cdot 9\right) \cdot t}\right)}}^{-1} \]

      5. fma-neg 82.3%

         \[\leadsto {\left(2 \cdot \frac{a}{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}\right)}^{-1} \]

      6. *-commutative 82.3%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, -\color{blue}{t \cdot \left(z \cdot 9\right)}\right)}\right)}^{-1} \]

      7. distribute-rgt-neg-in 82.3%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z \cdot 9\right)}\right)}\right)}^{-1} \]

      8. distribute-rgt-neg-in 82.3%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \color{blue}{\left(z \cdot \left(-9\right)\right)}\right)}\right)}^{-1} \]

      9. metadata-eval 82.3%

         \[\leadsto {\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot \color{blue}{-9}\right)\right)}\right)}^{-1} \]

   4. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 82.3%

         \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      2. associate-/r* 82.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}}} \]

      3. metadata-eval 82.3%

         \[\leadsto \frac{\color{blue}{0.5}}{\frac{a}{\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right)}} \]

      4. associate-*r* 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}} \]

      5. *-commutative 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      6. metadata-eval 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right)} \cdot \left(t \cdot z\right)\right)}} \]

      7. distribute-lft-neg-in 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9 \cdot \left(t \cdot z\right)}\right)}} \]

      8. distribute-lft-neg-in 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9\right) \cdot \left(t \cdot z\right)}\right)}} \]

      9. metadata-eval 82.3%

         \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{-9} \cdot \left(t \cdot z\right)\right)}} \]

      10. associate-*r* 82.3%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{\left(-9 \cdot t\right) \cdot z}\right)}} \]

      11. *-commutative 82.3%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}} \]

      12. *-commutative 82.3%

          \[\leadsto \frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot -9\right)}\right)}} \]

   6. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 83.2%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)} \]

      2. fma-undefine 83.2%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]

      3. distribute-lft-in 83.2%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right) + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)} \]

      4. *-commutative 83.2%

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a}} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right) \]

      5. associate-*l* 93.1%

         \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right) \]

   8. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a} + x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+235}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 1e+235)
      (/ (- (* x y) t_1) (* a_m 2.0))
      (* t (* -4.5 (/ z a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= 1e+235) {
		tmp = ((x * y) - t_1) / (a_m * 2.0);
	} else {
		tmp = t * (-4.5 * (z / a_m));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= 1d+235) then
        tmp = ((x * y) - t_1) / (a_m * 2.0d0)
    else
        tmp = t * ((-4.5d0) * (z / a_m))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= 1e+235) {
		tmp = ((x * y) - t_1) / (a_m * 2.0);
	} else {
		tmp = t * (-4.5 * (z / a_m));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= 1e+235:
		tmp = ((x * y) - t_1) / (a_m * 2.0)
	else:
		tmp = t * (-4.5 * (z / a_m))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= 1e+235)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a_m * 2.0));
	else
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= 1e+235)
		tmp = ((x * y) - t_1) / (a_m * 2.0);
	else
		tmp = t * (-4.5 * (z / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 1e+235], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.0000000000000001e235

   1. Initial program 94.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   if 1.0000000000000001e235 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 64.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.7%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 64.7%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 64.7%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 64.7%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 95.4%

         \[\leadsto \color{blue}{t \cdot \frac{z \cdot -9}{a \cdot 2}} \]

      2. *-commutative 95.4%

         \[\leadsto t \cdot \frac{\color{blue}{-9 \cdot z}}{a \cdot 2} \]

      3. *-commutative 95.4%

         \[\leadsto t \cdot \frac{-9 \cdot z}{\color{blue}{2 \cdot a}} \]

      4. times-frac 95.6%

         \[\leadsto t \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right)} \]

      5. metadata-eval 95.6%

         \[\leadsto t \cdot \left(\color{blue}{-4.5} \cdot \frac{z}{a}\right) \]

   7. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 10^{+235}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 6.5 \cdot 10^{+103}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 6.5e+103) (* -4.5 (/ (* z t) a_m)) (* t (* -4.5 (/ z a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 6.5e+103) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = t * (-4.5 * (z / a_m));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (a_m <= 6.5d+103) then
        tmp = (-4.5d0) * ((z * t) / a_m)
    else
        tmp = t * ((-4.5d0) * (z / a_m))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 6.5e+103) {
		tmp = -4.5 * ((z * t) / a_m);
	} else {
		tmp = t * (-4.5 * (z / a_m));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if a_m <= 6.5e+103:
		tmp = -4.5 * ((z * t) / a_m)
	else:
		tmp = t * (-4.5 * (z / a_m))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 6.5e+103)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a_m));
	else
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (a_m <= 6.5e+103)
		tmp = -4.5 * ((z * t) / a_m);
	else
		tmp = t * (-4.5 * (z / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 6.5e+103], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 6.50000000000000001e103

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 56.2%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   if 6.50000000000000001e103 < a

   1. Initial program 74.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.8%

      \[\leadsto \frac{\color{blue}{-9 \cdot \left(t \cdot z\right)}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. *-commutative 50.8%

         \[\leadsto \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      2. associate-*r* 50.6%

         \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]

   5. Simplified 50.6%

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 54.6%

         \[\leadsto \color{blue}{t \cdot \frac{z \cdot -9}{a \cdot 2}} \]

      2. *-commutative 54.6%

         \[\leadsto t \cdot \frac{\color{blue}{-9 \cdot z}}{a \cdot 2} \]

      3. *-commutative 54.6%

         \[\leadsto t \cdot \frac{-9 \cdot z}{\color{blue}{2 \cdot a}} \]

      4. times-frac 54.8%

         \[\leadsto t \cdot \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right)} \]

      5. metadata-eval 54.8%

         \[\leadsto t \cdot \left(\color{blue}{-4.5} \cdot \frac{z}{a}\right) \]

   7. Applied egg-rr 54.8%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{+103}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(-4.5 \cdot \frac{z \cdot t}{a\_m}\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* -4.5 (/ (* z t) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (-4.5 * ((z * t) / a_m));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    code = a_s * ((-4.5d0) * ((z * t) / a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (-4.5 * ((z * t) / a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (-4.5 * ((z * t) / a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(-4.5 * Float64(Float64(z * t) / a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (-4.5 * ((z * t) / a_m));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 55.3%

   \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

4. Final simplification 55.3%

   \[\leadsto -4.5 \cdot \frac{z \cdot t}{a} \]
5. Add Preprocessing

Developer target: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024085 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))