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

Percentage Accurate: 90.8% → 93.9%

Time: 14.2s

Alternatives: 11

Speedup: 0.5×

• 28.6% 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: 90.8% 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: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 4 \cdot 10^{+124}:\\ \;\;\;\;\frac{0.5}{\frac{a\_m}{\mathsf{fma}\left(t, z \cdot -9, x \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m \cdot 2} - \left(z \cdot 9\right) \cdot \frac{t}{a\_m \cdot 2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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) 4e+124)
    (/ 0.5 (/ a_m (fma t (* z -9.0) (* x y))))
    (- (* x (/ y (* a_m 2.0))) (* (* z 9.0) (/ t (* a_m 2.0)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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) <= 4e+124) {
		tmp = 0.5 / (a_m / fma(t, (z * -9.0), (x * y)));
	} else {
		tmp = (x * (y / (a_m * 2.0))) - ((z * 9.0) * (t / (a_m * 2.0)));
	}
	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])
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) <= 4e+124)
		tmp = Float64(0.5 / Float64(a_m / fma(t, Float64(z * -9.0), Float64(x * y))));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(a_m * 2.0))) - Float64(Float64(z * 9.0) * Float64(t / Float64(a_m * 2.0))));
	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.
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], 4e+124], N[(0.5 / N[(a$95$m / N[(t * N[(z * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * N[(t / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a 2) < 3.99999999999999979e124

   1. Initial program 88.8%

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

      1. clear-num 88.7%

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

      2. inv-pow 88.7%

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

      3. *-commutative 88.7%

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

      4. associate-/l* 88.7%

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

      5. fma-neg 88.7%

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

      6. *-commutative 88.7%

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

      7. distribute-rgt-neg-in 88.7%

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

      8. distribute-rgt-neg-in 88.7%

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

      9. metadata-eval 88.7%

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

   4. Applied egg-rr 88.7%

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

      1. unpow-1 88.7%

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

      2. associate-/r* 88.7%

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

      3. metadata-eval 88.7%

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

      4. fma-define 88.7%

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

      5. +-commutative 88.7%

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

      6. fma-define 89.6%

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

   6. Simplified 89.6%

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

   if 3.99999999999999979e124 < (*.f64 a 2)

   1. Initial program 69.7%

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

      1. div-sub 69.7%

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

      2. associate-/l* 76.4%

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

      3. associate-/l* 86.3%

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

   4. Applied egg-rr 86.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \leq 4 \cdot 10^{+124}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\mathsf{fma}\left(t, z \cdot -9, x \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2} - \left(z \cdot 9\right) \cdot \frac{t}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+291} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;x \cdot \frac{y}{a\_m \cdot 2} - \left(z \cdot 9\right) \cdot \frac{t}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a\_m \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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) (* t (* z 9.0)))))
   (*
    a_s
    (if (or (<= t_1 -5e+291) (not (<= t_1 INFINITY)))
      (- (* x (/ y (* a_m 2.0))) (* (* z 9.0) (/ t (* a_m 2.0))))
      (/ t_1 (* a_m 2.0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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) - (t * (z * 9.0));
	double tmp;
	if ((t_1 <= -5e+291) || !(t_1 <= ((double) INFINITY))) {
		tmp = (x * (y / (a_m * 2.0))) - ((z * 9.0) * (t / (a_m * 2.0)));
	} else {
		tmp = t_1 / (a_m * 2.0);
	}
	return a_s * tmp;
}

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
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) - (t * (z * 9.0));
	double tmp;
	if ((t_1 <= -5e+291) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (x * (y / (a_m * 2.0))) - ((z * 9.0) * (t / (a_m * 2.0)));
	} else {
		tmp = t_1 / (a_m * 2.0);
	}
	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])
[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) - (t * (z * 9.0))
	tmp = 0
	if (t_1 <= -5e+291) or not (t_1 <= math.inf):
		tmp = (x * (y / (a_m * 2.0))) - ((z * 9.0) * (t / (a_m * 2.0)))
	else:
		tmp = t_1 / (a_m * 2.0)
	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])
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(Float64(x * y) - Float64(t * Float64(z * 9.0)))
	tmp = 0.0
	if ((t_1 <= -5e+291) || !(t_1 <= Inf))
		tmp = Float64(Float64(x * Float64(y / Float64(a_m * 2.0))) - Float64(Float64(z * 9.0) * Float64(t / Float64(a_m * 2.0))));
	else
		tmp = Float64(t_1 / Float64(a_m * 2.0));
	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])){:}
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) - (t * (z * 9.0));
	tmp = 0.0;
	if ((t_1 <= -5e+291) || ~((t_1 <= Inf)))
		tmp = (x * (y / (a_m * 2.0))) - ((z * 9.0) * (t / (a_m * 2.0)));
	else
		tmp = t_1 / (a_m * 2.0);
	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.
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[(N[(x * y), $MachinePrecision] - N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, -5e+291], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(x * N[(y / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * N[(t / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -5.0000000000000001e291 or +inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 56.5%

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

      1. div-sub 50.0%

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

      2. associate-/l* 72.5%

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

      3. associate-/l* 84.7%

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

   4. Applied egg-rr 84.7%

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

   if -5.0000000000000001e291 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < +inf.0

   1. Initial program 92.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+291} \lor \neg \left(x \cdot y - t \cdot \left(z \cdot 9\right) \leq \infty\right):\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2} - \left(z \cdot 9\right) \cdot \frac{t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ t_2 := \frac{x \cdot y}{a\_m \cdot 2}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \frac{-4.5}{\frac{a\_m}{t}}\\ \mathbf{elif}\;x \cdot y \leq -200000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-124}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a\_m}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+240}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* 0.5 (/ y a_m)))) (t_2 (/ (* x y) (* a_m 2.0))))
   (*
    a_s
    (if (<= (* x y) -2e+94)
      t_1
      (if (<= (* x y) -1e+57)
        (* z (/ -4.5 (/ a_m t)))
        (if (<= (* x y) -200000000000.0)
          t_2
          (if (<= (* x y) 2e-124)
            (* t (/ -4.5 (/ a_m z)))
            (if (<= (* x y) 5e+240) t_2 t_1))))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 * (0.5 * (y / a_m));
	double t_2 = (x * y) / (a_m * 2.0);
	double tmp;
	if ((x * y) <= -2e+94) {
		tmp = t_1;
	} else if ((x * y) <= -1e+57) {
		tmp = z * (-4.5 / (a_m / t));
	} else if ((x * y) <= -200000000000.0) {
		tmp = t_2;
	} else if ((x * y) <= 2e-124) {
		tmp = t * (-4.5 / (a_m / z));
	} else if ((x * y) <= 5e+240) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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.
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) :: t_2
    real(8) :: tmp
    t_1 = x * (0.5d0 * (y / a_m))
    t_2 = (x * y) / (a_m * 2.0d0)
    if ((x * y) <= (-2d+94)) then
        tmp = t_1
    else if ((x * y) <= (-1d+57)) then
        tmp = z * ((-4.5d0) / (a_m / t))
    else if ((x * y) <= (-200000000000.0d0)) then
        tmp = t_2
    else if ((x * y) <= 2d-124) then
        tmp = t * ((-4.5d0) / (a_m / z))
    else if ((x * y) <= 5d+240) then
        tmp = t_2
    else
        tmp = t_1
    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;
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 * (0.5 * (y / a_m));
	double t_2 = (x * y) / (a_m * 2.0);
	double tmp;
	if ((x * y) <= -2e+94) {
		tmp = t_1;
	} else if ((x * y) <= -1e+57) {
		tmp = z * (-4.5 / (a_m / t));
	} else if ((x * y) <= -200000000000.0) {
		tmp = t_2;
	} else if ((x * y) <= 2e-124) {
		tmp = t * (-4.5 / (a_m / z));
	} else if ((x * y) <= 5e+240) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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])
[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 * (0.5 * (y / a_m))
	t_2 = (x * y) / (a_m * 2.0)
	tmp = 0
	if (x * y) <= -2e+94:
		tmp = t_1
	elif (x * y) <= -1e+57:
		tmp = z * (-4.5 / (a_m / t))
	elif (x * y) <= -200000000000.0:
		tmp = t_2
	elif (x * y) <= 2e-124:
		tmp = t * (-4.5 / (a_m / z))
	elif (x * y) <= 5e+240:
		tmp = t_2
	else:
		tmp = t_1
	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])
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(0.5 * Float64(y / a_m)))
	t_2 = Float64(Float64(x * y) / Float64(a_m * 2.0))
	tmp = 0.0
	if (Float64(x * y) <= -2e+94)
		tmp = t_1;
	elseif (Float64(x * y) <= -1e+57)
		tmp = Float64(z * Float64(-4.5 / Float64(a_m / t)));
	elseif (Float64(x * y) <= -200000000000.0)
		tmp = t_2;
	elseif (Float64(x * y) <= 2e-124)
		tmp = Float64(t * Float64(-4.5 / Float64(a_m / z)));
	elseif (Float64(x * y) <= 5e+240)
		tmp = t_2;
	else
		tmp = t_1;
	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])){:}
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 * (0.5 * (y / a_m));
	t_2 = (x * y) / (a_m * 2.0);
	tmp = 0.0;
	if ((x * y) <= -2e+94)
		tmp = t_1;
	elseif ((x * y) <= -1e+57)
		tmp = z * (-4.5 / (a_m / t));
	elseif ((x * y) <= -200000000000.0)
		tmp = t_2;
	elseif ((x * y) <= 2e-124)
		tmp = t * (-4.5 / (a_m / z));
	elseif ((x * y) <= 5e+240)
		tmp = t_2;
	else
		tmp = t_1;
	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.
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[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+94], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e+57], N[(z * N[(-4.5 / N[(a$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -200000000000.0], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 2e-124], N[(t * N[(-4.5 / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+240], t$95$2, t$95$1]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e94 or 5.0000000000000003e240 < (*.f64 x y)

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-/l* 93.4%

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

      3. associate-*l* 93.4%

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

   5. Simplified 93.4%

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

   if -2e94 < (*.f64 x y) < -1.00000000000000005e57

   1. Initial program 82.4%

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

   3. Taylor expanded in x around 0 73.4%

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

      1. associate-/l* 82.4%

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

   5. Simplified 82.4%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 82.4%

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

      2. associate-*r/ 73.4%

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

      3. metadata-eval 73.4%

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

      4. times-frac 73.3%

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

      5. associate-*r* 64.9%

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

      6. *-commutative 64.9%

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

      7. associate-*l* 73.6%

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

      8. clear-num 73.7%

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

      9. associate-*l* 64.9%

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

      10. *-commutative 64.9%

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

      11. associate-*r* 73.4%

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

      12. times-frac 73.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{t \cdot z} \cdot \frac{2}{-9}}} \]

      13. metadata-eval 73.3%

          \[\leadsto \frac{1}{\frac{a}{t \cdot z} \cdot \color{blue}{-0.2222222222222222}} \]

   7. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{t \cdot z} \cdot -0.2222222222222222}} \]
   8. Step-by-step derivation

      1. *-commutative 73.3%

         \[\leadsto \frac{1}{\color{blue}{-0.2222222222222222 \cdot \frac{a}{t \cdot z}}} \]

      2. associate-/r* 73.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{-0.2222222222222222}}{\frac{a}{t \cdot z}}} \]

      3. metadata-eval 73.3%

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

   9. Simplified 73.3%

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

       1. associate-/r* 73.8%

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

       2. associate-/r/ 73.8%

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

   11. Applied egg-rr 73.8%

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

   if -1.00000000000000005e57 < (*.f64 x y) < -2e11 or 1.99999999999999987e-124 < (*.f64 x y) < 5.0000000000000003e240

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf 74.3%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   if -2e11 < (*.f64 x y) < 1.99999999999999987e-124

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0 77.0%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

   5. Simplified 76.9%

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

      1. associate-*l* 76.9%

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

      2. *-commutative 76.9%

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

      3. associate-*r* 77.0%

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

      4. times-frac 77.0%

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

      5. associate-*r/ 77.2%

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

      6. metadata-eval 77.2%

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

      7. *-commutative 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-*r* 77.3%

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

   7. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
   8. Step-by-step derivation

      1. clear-num 77.0%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{-4.5}{\frac{a}{z}}} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \frac{-4.5}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y \leq -200000000000:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-124}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+44}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a\_m}\\ \mathbf{elif}\;x \leq -0.088:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= x -5.5e+92)
      t_1
      (if (<= x -1.3e+44)
        (* -4.5 (/ (* t z) a_m))
        (if (<= x -0.088)
          t_1
          (if (<= x 1.25e-226)
            (* t (* -4.5 (/ z a_m)))
            (* x (* 0.5 (/ y a_m))))))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -5.5e+92) {
		tmp = t_1;
	} else if (x <= -1.3e+44) {
		tmp = -4.5 * ((t * z) / a_m);
	} else if (x <= -0.088) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 * (z / a_m));
	} else {
		tmp = x * (0.5 * (y / 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.
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 = (0.5d0 * y) * (x / a_m)
    if (x <= (-5.5d+92)) then
        tmp = t_1
    else if (x <= (-1.3d+44)) then
        tmp = (-4.5d0) * ((t * z) / a_m)
    else if (x <= (-0.088d0)) then
        tmp = t_1
    else if (x <= 1.25d-226) then
        tmp = t * ((-4.5d0) * (z / a_m))
    else
        tmp = x * (0.5d0 * (y / 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;
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -5.5e+92) {
		tmp = t_1;
	} else if (x <= -1.3e+44) {
		tmp = -4.5 * ((t * z) / a_m);
	} else if (x <= -0.088) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 * (z / a_m));
	} else {
		tmp = x * (0.5 * (y / 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])
[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 = (0.5 * y) * (x / a_m)
	tmp = 0
	if x <= -5.5e+92:
		tmp = t_1
	elif x <= -1.3e+44:
		tmp = -4.5 * ((t * z) / a_m)
	elif x <= -0.088:
		tmp = t_1
	elif x <= 1.25e-226:
		tmp = t * (-4.5 * (z / a_m))
	else:
		tmp = x * (0.5 * (y / 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])
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(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (x <= -5.5e+92)
		tmp = t_1;
	elseif (x <= -1.3e+44)
		tmp = Float64(-4.5 * Float64(Float64(t * z) / a_m));
	elseif (x <= -0.088)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = Float64(t * Float64(-4.5 * Float64(z / a_m)));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y / 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])){:}
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 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if (x <= -5.5e+92)
		tmp = t_1;
	elseif (x <= -1.3e+44)
		tmp = -4.5 * ((t * z) / a_m);
	elseif (x <= -0.088)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = t * (-4.5 * (z / a_m));
	else
		tmp = x * (0.5 * (y / 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.
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[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -5.5e+92], t$95$1, If[LessEqual[x, -1.3e+44], N[(-4.5 * N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.088], t$95$1, If[LessEqual[x, 1.25e-226], N[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -5.50000000000000053e92 or -1.3e44 < x < -0.087999999999999995

   1. Initial program 84.7%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-/l* 84.5%

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

      5. fma-neg 84.5%

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

      6. *-commutative 84.5%

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

      7. distribute-rgt-neg-in 84.5%

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

      8. distribute-rgt-neg-in 84.5%

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

      9. metadata-eval 84.5%

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

   4. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/r* 84.5%

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

      3. metadata-eval 84.5%

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

      4. fma-define 84.5%

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

      5. +-commutative 84.5%

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

      6. fma-define 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in t around 0 78.9%

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

      1. associate-/r* 90.1%

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

   9. Simplified 90.1%

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

       1. associate-/r/ 90.0%

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

       2. *-commutative 90.0%

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

       3. div-inv 90.0%

          \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{a}{x}}\right)} \]

       4. clear-num 90.0%

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

   11. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{a}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 90.0%

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

   13. Simplified 90.0%

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

   if -5.50000000000000053e92 < x < -1.3e44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.3%

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

   if -0.087999999999999995 < x < 1.2499999999999999e-226

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. associate-/l* 68.3%

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

      3. associate-*l* 68.4%

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

   5. Simplified 68.4%

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

   if 1.2499999999999999e-226 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in x around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/l* 60.2%

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

      3. associate-*l* 60.2%

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

   5. Simplified 60.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+92}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+44}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \leq -0.088:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+43}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a\_m}\\ \mathbf{elif}\;x \leq -0.00062:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= x -4.8e+92)
      t_1
      (if (<= x -6e+43)
        (* -4.5 (/ (* t z) a_m))
        (if (<= x -0.00062)
          t_1
          (if (<= x 1.25e-226)
            (* t (/ -4.5 (/ a_m z)))
            (* x (* 0.5 (/ y a_m))))))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -4.8e+92) {
		tmp = t_1;
	} else if (x <= -6e+43) {
		tmp = -4.5 * ((t * z) / a_m);
	} else if (x <= -0.00062) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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.
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 = (0.5d0 * y) * (x / a_m)
    if (x <= (-4.8d+92)) then
        tmp = t_1
    else if (x <= (-6d+43)) then
        tmp = (-4.5d0) * ((t * z) / a_m)
    else if (x <= (-0.00062d0)) then
        tmp = t_1
    else if (x <= 1.25d-226) then
        tmp = t * ((-4.5d0) / (a_m / z))
    else
        tmp = x * (0.5d0 * (y / 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;
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -4.8e+92) {
		tmp = t_1;
	} else if (x <= -6e+43) {
		tmp = -4.5 * ((t * z) / a_m);
	} else if (x <= -0.00062) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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])
[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 = (0.5 * y) * (x / a_m)
	tmp = 0
	if x <= -4.8e+92:
		tmp = t_1
	elif x <= -6e+43:
		tmp = -4.5 * ((t * z) / a_m)
	elif x <= -0.00062:
		tmp = t_1
	elif x <= 1.25e-226:
		tmp = t * (-4.5 / (a_m / z))
	else:
		tmp = x * (0.5 * (y / 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])
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(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (x <= -4.8e+92)
		tmp = t_1;
	elseif (x <= -6e+43)
		tmp = Float64(-4.5 * Float64(Float64(t * z) / a_m));
	elseif (x <= -0.00062)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = Float64(t * Float64(-4.5 / Float64(a_m / z)));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y / 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])){:}
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 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if (x <= -4.8e+92)
		tmp = t_1;
	elseif (x <= -6e+43)
		tmp = -4.5 * ((t * z) / a_m);
	elseif (x <= -0.00062)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = t * (-4.5 / (a_m / z));
	else
		tmp = x * (0.5 * (y / 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.
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[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -4.8e+92], t$95$1, If[LessEqual[x, -6e+43], N[(-4.5 * N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.00062], t$95$1, If[LessEqual[x, 1.25e-226], N[(t * N[(-4.5 / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -4.80000000000000009e92 or -6.00000000000000033e43 < x < -6.2e-4

   1. Initial program 84.7%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-/l* 84.5%

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

      5. fma-neg 84.5%

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

      6. *-commutative 84.5%

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

      7. distribute-rgt-neg-in 84.5%

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

      8. distribute-rgt-neg-in 84.5%

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

      9. metadata-eval 84.5%

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

   4. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/r* 84.5%

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

      3. metadata-eval 84.5%

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

      4. fma-define 84.5%

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

      5. +-commutative 84.5%

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

      6. fma-define 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in t around 0 78.9%

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

      1. associate-/r* 90.1%

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

   9. Simplified 90.1%

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

       1. associate-/r/ 90.0%

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

       2. *-commutative 90.0%

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

       3. div-inv 90.0%

          \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{a}{x}}\right)} \]

       4. clear-num 90.0%

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

   11. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{a}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 90.0%

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

   13. Simplified 90.0%

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

   if -4.80000000000000009e92 < x < -6.00000000000000033e43

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.3%

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

   if -6.2e-4 < x < 1.2499999999999999e-226

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

   5. Simplified 63.4%

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

      1. associate-*l* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. times-frac 63.5%

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

      5. associate-*r/ 68.3%

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

      6. metadata-eval 68.3%

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

      7. *-commutative 68.3%

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

      8. *-commutative 68.3%

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

      9. associate-*r* 68.4%

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

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
   8. Step-by-step derivation

      1. clear-num 68.1%

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

      2. un-div-inv 68.2%

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

   9. Applied egg-rr 68.2%

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

   if 1.2499999999999999e-226 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in x around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/l* 60.2%

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

      3. associate-*l* 60.2%

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

   5. Simplified 60.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+43}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \leq -0.00062:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-4.5}{\frac{a\_m}{t \cdot z}}\\ \mathbf{elif}\;x \leq -0.19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* (* 0.5 y) (/ x a_m))))
   (*
    a_s
    (if (<= x -4.8e+92)
      t_1
      (if (<= x -1.05e+44)
        (/ -4.5 (/ a_m (* t z)))
        (if (<= x -0.19)
          t_1
          (if (<= x 1.25e-226)
            (* t (/ -4.5 (/ a_m z)))
            (* x (* 0.5 (/ y a_m))))))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -4.8e+92) {
		tmp = t_1;
	} else if (x <= -1.05e+44) {
		tmp = -4.5 / (a_m / (t * z));
	} else if (x <= -0.19) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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.
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 = (0.5d0 * y) * (x / a_m)
    if (x <= (-4.8d+92)) then
        tmp = t_1
    else if (x <= (-1.05d+44)) then
        tmp = (-4.5d0) / (a_m / (t * z))
    else if (x <= (-0.19d0)) then
        tmp = t_1
    else if (x <= 1.25d-226) then
        tmp = t * ((-4.5d0) / (a_m / z))
    else
        tmp = x * (0.5d0 * (y / 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;
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 = (0.5 * y) * (x / a_m);
	double tmp;
	if (x <= -4.8e+92) {
		tmp = t_1;
	} else if (x <= -1.05e+44) {
		tmp = -4.5 / (a_m / (t * z));
	} else if (x <= -0.19) {
		tmp = t_1;
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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])
[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 = (0.5 * y) * (x / a_m)
	tmp = 0
	if x <= -4.8e+92:
		tmp = t_1
	elif x <= -1.05e+44:
		tmp = -4.5 / (a_m / (t * z))
	elif x <= -0.19:
		tmp = t_1
	elif x <= 1.25e-226:
		tmp = t * (-4.5 / (a_m / z))
	else:
		tmp = x * (0.5 * (y / 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])
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(Float64(0.5 * y) * Float64(x / a_m))
	tmp = 0.0
	if (x <= -4.8e+92)
		tmp = t_1;
	elseif (x <= -1.05e+44)
		tmp = Float64(-4.5 / Float64(a_m / Float64(t * z)));
	elseif (x <= -0.19)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = Float64(t * Float64(-4.5 / Float64(a_m / z)));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y / 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])){:}
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 = (0.5 * y) * (x / a_m);
	tmp = 0.0;
	if (x <= -4.8e+92)
		tmp = t_1;
	elseif (x <= -1.05e+44)
		tmp = -4.5 / (a_m / (t * z));
	elseif (x <= -0.19)
		tmp = t_1;
	elseif (x <= 1.25e-226)
		tmp = t * (-4.5 / (a_m / z));
	else
		tmp = x * (0.5 * (y / 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.
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[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[x, -4.8e+92], t$95$1, If[LessEqual[x, -1.05e+44], N[(-4.5 / N[(a$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.19], t$95$1, If[LessEqual[x, 1.25e-226], N[(t * N[(-4.5 / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < -4.80000000000000009e92 or -1.04999999999999993e44 < x < -0.19

   1. Initial program 84.7%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

      3. *-commutative 84.5%

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

      4. associate-/l* 84.5%

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

      5. fma-neg 84.5%

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

      6. *-commutative 84.5%

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

      7. distribute-rgt-neg-in 84.5%

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

      8. distribute-rgt-neg-in 84.5%

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

      9. metadata-eval 84.5%

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

   4. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/r* 84.5%

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

      3. metadata-eval 84.5%

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

      4. fma-define 84.5%

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

      5. +-commutative 84.5%

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

      6. fma-define 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in t around 0 78.9%

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

      1. associate-/r* 90.1%

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

   9. Simplified 90.1%

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

       1. associate-/r/ 90.0%

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

       2. *-commutative 90.0%

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

       3. div-inv 90.0%

          \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{a}{x}}\right)} \]

       4. clear-num 90.0%

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

   11. Applied egg-rr 90.0%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{a}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 90.0%

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

   13. Simplified 90.0%

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

   if -4.80000000000000009e92 < x < -1.04999999999999993e44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. associate-/l* 56.8%

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

   5. Simplified 56.8%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 56.8%

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

      2. associate-*r/ 67.3%

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

      3. metadata-eval 67.3%

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

      4. times-frac 67.3%

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

      5. associate-*r* 67.6%

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

      6. *-commutative 67.6%

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

      7. associate-*l* 67.5%

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

      8. clear-num 67.6%

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

      9. associate-*l* 67.5%

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

      10. *-commutative 67.5%

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

      11. associate-*r* 67.5%

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

      12. times-frac 67.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{t \cdot z} \cdot \frac{2}{-9}}} \]

      13. metadata-eval 67.5%

          \[\leadsto \frac{1}{\frac{a}{t \cdot z} \cdot \color{blue}{-0.2222222222222222}} \]

   7. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{t \cdot z} \cdot -0.2222222222222222}} \]
   8. Step-by-step derivation

      1. *-commutative 67.5%

         \[\leadsto \frac{1}{\color{blue}{-0.2222222222222222 \cdot \frac{a}{t \cdot z}}} \]

      2. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{-0.2222222222222222}}{\frac{a}{t \cdot z}}} \]

      3. metadata-eval 67.5%

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

   9. Simplified 67.5%

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

   if -0.19 < x < 1.2499999999999999e-226

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

   5. Simplified 63.4%

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

      1. associate-*l* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. times-frac 63.5%

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

      5. associate-*r/ 68.3%

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

      6. metadata-eval 68.3%

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

      7. *-commutative 68.3%

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

      8. *-commutative 68.3%

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

      9. associate-*r* 68.4%

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

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
   8. Step-by-step derivation

      1. clear-num 68.1%

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

      2. un-div-inv 68.2%

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

   9. Applied egg-rr 68.2%

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

   if 1.2499999999999999e-226 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in x around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/l* 60.2%

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

      3. associate-*l* 60.2%

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

   5. Simplified 60.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;x \leq -0.19:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a\_m}{x}}{y}}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+44}:\\ \;\;\;\;\frac{-4.5}{\frac{a\_m}{t \cdot z}}\\ \mathbf{elif}\;x \leq -0.0026:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 -4.8e+92)
    (/ 0.5 (/ (/ a_m x) y))
    (if (<= x -1.25e+44)
      (/ -4.5 (/ a_m (* t z)))
      (if (<= x -0.0026)
        (* (* 0.5 y) (/ x a_m))
        (if (<= x 1.25e-226)
          (* t (/ -4.5 (/ a_m z)))
          (* x (* 0.5 (/ y a_m)))))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 <= -4.8e+92) {
		tmp = 0.5 / ((a_m / x) / y);
	} else if (x <= -1.25e+44) {
		tmp = -4.5 / (a_m / (t * z));
	} else if (x <= -0.0026) {
		tmp = (0.5 * y) * (x / a_m);
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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.
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 <= (-4.8d+92)) then
        tmp = 0.5d0 / ((a_m / x) / y)
    else if (x <= (-1.25d+44)) then
        tmp = (-4.5d0) / (a_m / (t * z))
    else if (x <= (-0.0026d0)) then
        tmp = (0.5d0 * y) * (x / a_m)
    else if (x <= 1.25d-226) then
        tmp = t * ((-4.5d0) / (a_m / z))
    else
        tmp = x * (0.5d0 * (y / 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;
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 <= -4.8e+92) {
		tmp = 0.5 / ((a_m / x) / y);
	} else if (x <= -1.25e+44) {
		tmp = -4.5 / (a_m / (t * z));
	} else if (x <= -0.0026) {
		tmp = (0.5 * y) * (x / a_m);
	} else if (x <= 1.25e-226) {
		tmp = t * (-4.5 / (a_m / z));
	} else {
		tmp = x * (0.5 * (y / 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])
[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 <= -4.8e+92:
		tmp = 0.5 / ((a_m / x) / y)
	elif x <= -1.25e+44:
		tmp = -4.5 / (a_m / (t * z))
	elif x <= -0.0026:
		tmp = (0.5 * y) * (x / a_m)
	elif x <= 1.25e-226:
		tmp = t * (-4.5 / (a_m / z))
	else:
		tmp = x * (0.5 * (y / 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])
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 (x <= -4.8e+92)
		tmp = Float64(0.5 / Float64(Float64(a_m / x) / y));
	elseif (x <= -1.25e+44)
		tmp = Float64(-4.5 / Float64(a_m / Float64(t * z)));
	elseif (x <= -0.0026)
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	elseif (x <= 1.25e-226)
		tmp = Float64(t * Float64(-4.5 / Float64(a_m / z)));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y / 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])){:}
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 <= -4.8e+92)
		tmp = 0.5 / ((a_m / x) / y);
	elseif (x <= -1.25e+44)
		tmp = -4.5 / (a_m / (t * z));
	elseif (x <= -0.0026)
		tmp = (0.5 * y) * (x / a_m);
	elseif (x <= 1.25e-226)
		tmp = t * (-4.5 / (a_m / z));
	else
		tmp = x * (0.5 * (y / 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.
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[x, -4.8e+92], N[(0.5 / N[(N[(a$95$m / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.25e+44], N[(-4.5 / N[(a$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.0026], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-226], N[(t * N[(-4.5 / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if x < -4.80000000000000009e92

   1. Initial program 84.5%

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

      1. clear-num 84.3%

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

      2. inv-pow 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-/l* 84.3%

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

      5. fma-neg 84.3%

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

      6. *-commutative 84.3%

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

      7. distribute-rgt-neg-in 84.3%

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

      8. distribute-rgt-neg-in 84.3%

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

      9. metadata-eval 84.3%

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

   4. Applied egg-rr 84.3%

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

      1. unpow-1 84.3%

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

      2. associate-/r* 84.3%

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

      3. metadata-eval 84.3%

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

      4. fma-define 84.3%

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

      5. +-commutative 84.3%

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

      6. fma-define 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. associate-/r* 88.6%

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

   9. Simplified 88.6%

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

   if -4.80000000000000009e92 < x < -1.2499999999999999e44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. associate-/l* 56.8%

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

   5. Simplified 56.8%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 56.8%

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

      2. associate-*r/ 67.3%

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

      3. metadata-eval 67.3%

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

      4. times-frac 67.3%

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

      5. associate-*r* 67.6%

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

      6. *-commutative 67.6%

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

      7. associate-*l* 67.5%

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

      8. clear-num 67.6%

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

      9. associate-*l* 67.5%

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

      10. *-commutative 67.5%

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

      11. associate-*r* 67.5%

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

      12. times-frac 67.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{t \cdot z} \cdot \frac{2}{-9}}} \]

      13. metadata-eval 67.5%

          \[\leadsto \frac{1}{\frac{a}{t \cdot z} \cdot \color{blue}{-0.2222222222222222}} \]

   7. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{t \cdot z} \cdot -0.2222222222222222}} \]
   8. Step-by-step derivation

      1. *-commutative 67.5%

         \[\leadsto \frac{1}{\color{blue}{-0.2222222222222222 \cdot \frac{a}{t \cdot z}}} \]

      2. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{-0.2222222222222222}}{\frac{a}{t \cdot z}}} \]

      3. metadata-eval 67.5%

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

   9. Simplified 67.5%

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

   if -1.2499999999999999e44 < x < -0.0025999999999999999

   1. Initial program 85.8%

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

      1. clear-num 85.6%

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

      2. inv-pow 85.6%

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

      3. *-commutative 85.6%

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

      4. associate-/l* 85.6%

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

      5. fma-neg 85.6%

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

      6. *-commutative 85.6%

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

      7. distribute-rgt-neg-in 85.6%

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

      8. distribute-rgt-neg-in 85.6%

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

      9. metadata-eval 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. unpow-1 85.6%

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

      2. associate-/r* 85.6%

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

      3. metadata-eval 85.6%

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

      4. fma-define 85.6%

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

      5. +-commutative 85.6%

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

      6. fma-define 85.6%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in t around 0 85.6%

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

      1. associate-/r* 99.6%

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

   9. Simplified 99.6%

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

       1. associate-/r/ 99.8%

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

       2. *-commutative 99.8%

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

       3. div-inv 99.8%

          \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{1}{\frac{a}{x}}\right)} \]

       4. clear-num 99.6%

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

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \frac{x}{a}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 99.6%

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

   13. Simplified 99.6%

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

   if -0.0025999999999999999 < x < 1.2499999999999999e-226

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

   5. Simplified 63.4%

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

      1. associate-*l* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. times-frac 63.5%

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

      5. associate-*r/ 68.3%

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

      6. metadata-eval 68.3%

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

      7. *-commutative 68.3%

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

      8. *-commutative 68.3%

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

      9. associate-*r* 68.4%

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

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
   8. Step-by-step derivation

      1. clear-num 68.1%

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

      2. un-div-inv 68.2%

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

   9. Applied egg-rr 68.2%

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

   if 1.2499999999999999e-226 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in x around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/l* 60.2%

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

      3. associate-*l* 60.2%

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

   5. Simplified 60.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{x}}{y}}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+44}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{t \cdot z}}\\ \mathbf{elif}\;x \leq -0.0026:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+265} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+248}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a\_m \cdot 2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (or (<= (* x y) -5e+265) (not (<= (* x y) 5e+248)))
    (* x (* 0.5 (/ y a_m)))
    (/ (- (* x y) (* t (* z 9.0))) (* a_m 2.0)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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+265) || !((x * y) <= 5e+248)) {
		tmp = x * (0.5 * (y / a_m));
	} else {
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	}
	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.
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+265)) .or. (.not. ((x * y) <= 5d+248))) then
        tmp = x * (0.5d0 * (y / a_m))
    else
        tmp = ((x * y) - (t * (z * 9.0d0))) / (a_m * 2.0d0)
    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;
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+265) || !((x * y) <= 5e+248)) {
		tmp = x * (0.5 * (y / a_m));
	} else {
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	}
	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])
[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+265) or not ((x * y) <= 5e+248):
		tmp = x * (0.5 * (y / a_m))
	else:
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0)
	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])
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+265) || !(Float64(x * y) <= 5e+248))
		tmp = Float64(x * Float64(0.5 * Float64(y / a_m)));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(t * Float64(z * 9.0))) / Float64(a_m * 2.0));
	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])){:}
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+265) || ~(((x * y) <= 5e+248)))
		tmp = x * (0.5 * (y / a_m));
	else
		tmp = ((x * y) - (t * (z * 9.0))) / (a_m * 2.0);
	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.
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[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+265], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+248]], $MachinePrecision]], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000002e265 or 4.9999999999999996e248 < (*.f64 x y)

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-/l* 98.1%

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

      3. associate-*l* 98.1%

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

   5. Simplified 98.1%

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

   if -5.0000000000000002e265 < (*.f64 x y) < 4.9999999999999996e248

   1. Initial program 92.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+265} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+248}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t \cdot \left(z \cdot 9\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-117} \lor \neg \left(y \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (or (<= y -2.05e-117) (not (<= y 1.7e+54)))
    (* x (* 0.5 (/ y 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);
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 ((y <= -2.05e-117) || !(y <= 1.7e+54)) {
		tmp = x * (0.5 * (y / 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.
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 ((y <= (-2.05d-117)) .or. (.not. (y <= 1.7d+54))) then
        tmp = x * (0.5d0 * (y / 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;
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 ((y <= -2.05e-117) || !(y <= 1.7e+54)) {
		tmp = x * (0.5 * (y / 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])
[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 (y <= -2.05e-117) or not (y <= 1.7e+54):
		tmp = x * (0.5 * (y / 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])
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 ((y <= -2.05e-117) || !(y <= 1.7e+54))
		tmp = Float64(x * Float64(0.5 * Float64(y / 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])){:}
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 ((y <= -2.05e-117) || ~((y <= 1.7e+54)))
		tmp = x * (0.5 * (y / 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.
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[Or[LessEqual[y, -2.05e-117], N[Not[LessEqual[y, 1.7e+54]], $MachinePrecision]], N[(x * N[(0.5 * N[(y / a$95$m), $MachinePrecision]), $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 y < -2.05000000000000016e-117 or 1.7e54 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in x around inf 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-/l* 70.7%

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

      3. associate-*l* 70.7%

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

   5. Simplified 70.7%

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

   if -2.05000000000000016e-117 < y < 1.7e54

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-/l* 64.6%

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

      3. associate-*l* 64.7%

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

   5. Simplified 64.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-117} \lor \neg \left(y \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% accurate, 1.9× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* t (/ z a_m)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
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 * (t * (z / 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.
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) * (t * (z / 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;
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 * (t * (z / 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])
[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 * (t * (z / 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])
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(t * Float64(z / 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])){:}
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 * (t * (z / 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.
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[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in x around 0 42.8%

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

   1. associate-/l* 46.2%

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

5. Simplified 46.2%

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

6. Final simplification 46.2%

   \[\leadsto -4.5 \cdot \left(t \cdot \frac{z}{a}\right) \]
7. Add Preprocessing

Alternative 11: 51.4% accurate, 1.9× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* 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);
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 * (t * (-4.5 * (z / 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.
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 * (t * ((-4.5d0) * (z / 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;
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 * (t * (-4.5 * (z / 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])
[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 * (t * (-4.5 * (z / 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])
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(t * Float64(-4.5 * Float64(z / 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])){:}
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 * (t * (-4.5 * (z / 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.
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[(t * N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in x around 0 42.8%

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

   1. *-commutative 42.8%

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

   2. associate-/l* 46.2%

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

   3. associate-*l* 46.2%

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

5. Simplified 46.2%

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

6. Final simplification 46.2%

   \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a}\right) \]
7. Add Preprocessing

Developer target: 94.0% 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 2024041 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (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)))