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

Percentage Accurate: 91.4% → 96.0%

Time: 14.2s

Alternatives: 11

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% 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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+295} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+295}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \left(x \cdot \frac{y}{t \cdot a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -5e+295) (not (<= t_1 2e+295)))
     (* t (+ (* -4.5 (/ z a)) (* 0.5 (* x (/ y (* t a))))))
     (/ t_1 (* 2.0 a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -5e+295) || !(t_1 <= 2e+295)) {
		tmp = t * ((-4.5 * (z / a)) + (0.5 * (x * (y / (t * a)))));
	} else {
		tmp = t_1 / (2.0 * a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - ((z * 9.0d0) * t)
    if ((t_1 <= (-5d+295)) .or. (.not. (t_1 <= 2d+295))) then
        tmp = t * (((-4.5d0) * (z / a)) + (0.5d0 * (x * (y / (t * a)))))
    else
        tmp = t_1 / (2.0d0 * a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -5e+295) || !(t_1 <= 2e+295)) {
		tmp = t * ((-4.5 * (z / a)) + (0.5 * (x * (y / (t * a)))));
	} else {
		tmp = t_1 / (2.0 * a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - ((z * 9.0) * t)
	tmp = 0
	if (t_1 <= -5e+295) or not (t_1 <= 2e+295):
		tmp = t * ((-4.5 * (z / a)) + (0.5 * (x * (y / (t * a)))))
	else:
		tmp = t_1 / (2.0 * a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -5e+295) || !(t_1 <= 2e+295))
		tmp = Float64(t * Float64(Float64(-4.5 * Float64(z / a)) + Float64(0.5 * Float64(x * Float64(y / Float64(t * a))))));
	else
		tmp = Float64(t_1 / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - ((z * 9.0) * t);
	tmp = 0.0;
	if ((t_1 <= -5e+295) || ~((t_1 <= 2e+295)))
		tmp = t * ((-4.5 * (z / a)) + (0.5 * (x * (y / (t * a)))));
	else
		tmp = t_1 / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+295], N[Not[LessEqual[t$95$1, 2e+295]], $MachinePrecision]], N[(t * N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(y / N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -4.99999999999999991e295 or 2e295 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 72.0%

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

   3. Taylor expanded in t around inf 83.9%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 88.1%

         \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{y}{a \cdot t}\right)}\right) \]

      2. *-commutative 88.1%

         \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \left(x \cdot \frac{y}{\color{blue}{t \cdot a}}\right)\right) \]

   5. Applied egg-rr 88.1%

      \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{y}{t \cdot a}\right)}\right) \]

   if -4.99999999999999991e295 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e295

   1. Initial program 97.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+295} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+295}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \left(x \cdot \frac{y}{t \cdot a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{2}, t \cdot \left(z \cdot -4.5\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) 5e+277)
   (/ (fma x (/ y 2.0) (* t (* z -4.5))) a)
   (* -4.5 (* z (/ t a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= 5e+277) {
		tmp = fma(x, (y / 2.0), (t * (z * -4.5))) / a;
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= 5e+277)
		tmp = Float64(fma(x, Float64(y / 2.0), Float64(t * Float64(z * -4.5))) / a);
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], 5e+277], N[(N[(x * N[(y / 2.0), $MachinePrecision] + N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999982e277

   1. Initial program 94.0%

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

      1. associate-/l/ 94.4%

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

      2. div-sub 94.4%

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

      3. associate-/l* 94.4%

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

      4. fma-neg 94.4%

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

      5. *-commutative 94.4%

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

      6. associate-/l* 94.4%

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

      7. distribute-rgt-neg-out 94.4%

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

      8. distribute-frac-neg 94.4%

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

      9. distribute-rgt-neg-in 94.4%

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

      10. associate-/l* 94.4%

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

      11. metadata-eval 94.4%

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

      12. metadata-eval 94.4%

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{y}{2}, t \cdot \left(z \cdot \color{blue}{-4.5}\right)\right)}{a} \]

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{y}{2}, t \cdot \left(z \cdot -4.5\right)\right)}{a}} \]
   4. Add Preprocessing

   if 4.99999999999999982e277 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 59.6%

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

   3. Taylor expanded in y around inf 59.0%

      \[\leadsto \color{blue}{y \cdot \left(-4.5 \cdot \frac{t \cdot z}{a \cdot y} + 0.5 \cdot \frac{x}{a}\right)} \]

   4. Taylor expanded in y around 0 67.1%

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

      1. associate-*l/ 96.0%

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

      2. *-commutative 96.0%

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

   6. Simplified 96.0%

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

Alternative 3: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) 5e+277)
   (* (fma z (* t -9.0) (* x y)) (/ 0.5 a))
   (* -4.5 (* z (/ t a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= 5e+277) {
		tmp = fma(z, (t * -9.0), (x * y)) * (0.5 / a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= 5e+277)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * Float64(0.5 / a));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], 5e+277], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999982e277

   1. Initial program 94.0%

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

   3. Taylor expanded in a around 0 94.4%

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

      1. associate-*r/ 94.4%

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

      2. cancel-sign-sub-inv 94.4%

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

      3. metadata-eval 94.4%

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

      4. *-commutative 94.4%

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

      5. *-commutative 94.4%

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

      6. associate-*r* 94.4%

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

      7. fma-define 94.4%

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

      8. associate-*l/ 94.3%

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

      9. *-commutative 94.3%

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

      10. fma-define 94.3%

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

      11. +-commutative 94.3%

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

      12. fma-define 94.3%

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

   5. Simplified 94.3%

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

   if 4.99999999999999982e277 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 59.6%

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

   3. Taylor expanded in y around inf 59.0%

      \[\leadsto \color{blue}{y \cdot \left(-4.5 \cdot \frac{t \cdot z}{a \cdot y} + 0.5 \cdot \frac{x}{a}\right)} \]

   4. Taylor expanded in y around 0 67.1%

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

      1. associate-*l/ 96.0%

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

      2. *-commutative 96.0%

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

   6. Simplified 96.0%

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

Alternative 4: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-48} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* x y) (/ 0.5 a))))
   (if (<= (* x y) -1e-126)
     t_1
     (if (<= (* x y) 4e-95)
       (* t (* z (/ -4.5 a)))
       (if (or (<= (* x y) 2e-48) (not (<= (* x y) 5e-11)))
         t_1
         (* -4.5 (/ (* z t) a)))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) * (0.5 / a);
	double tmp;
	if ((x * y) <= -1e-126) {
		tmp = t_1;
	} else if ((x * y) <= 4e-95) {
		tmp = t * (z * (-4.5 / a));
	} else if (((x * y) <= 2e-48) || !((x * y) <= 5e-11)) {
		tmp = t_1;
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) * (0.5d0 / a)
    if ((x * y) <= (-1d-126)) then
        tmp = t_1
    else if ((x * y) <= 4d-95) then
        tmp = t * (z * ((-4.5d0) / a))
    else if (((x * y) <= 2d-48) .or. (.not. ((x * y) <= 5d-11))) then
        tmp = t_1
    else
        tmp = (-4.5d0) * ((z * t) / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) * (0.5 / a);
	double tmp;
	if ((x * y) <= -1e-126) {
		tmp = t_1;
	} else if ((x * y) <= 4e-95) {
		tmp = t * (z * (-4.5 / a));
	} else if (((x * y) <= 2e-48) || !((x * y) <= 5e-11)) {
		tmp = t_1;
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) * (0.5 / a)
	tmp = 0
	if (x * y) <= -1e-126:
		tmp = t_1
	elif (x * y) <= 4e-95:
		tmp = t * (z * (-4.5 / a))
	elif ((x * y) <= 2e-48) or not ((x * y) <= 5e-11):
		tmp = t_1
	else:
		tmp = -4.5 * ((z * t) / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) * Float64(0.5 / a))
	tmp = 0.0
	if (Float64(x * y) <= -1e-126)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-95)
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	elseif ((Float64(x * y) <= 2e-48) || !(Float64(x * y) <= 5e-11))
		tmp = t_1;
	else
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) * (0.5 / a);
	tmp = 0.0;
	if ((x * y) <= -1e-126)
		tmp = t_1;
	elseif ((x * y) <= 4e-95)
		tmp = t * (z * (-4.5 / a));
	elseif (((x * y) <= 2e-48) || ~(((x * y) <= 5e-11)))
		tmp = t_1;
	else
		tmp = -4.5 * ((z * t) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e-126], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-95], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], 2e-48], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-11]], $MachinePrecision]], t$95$1, N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999995e-127 or 3.99999999999999996e-95 < (*.f64 x y) < 1.9999999999999999e-48 or 5.00000000000000018e-11 < (*.f64 x y)

   1. Initial program 92.3%

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

   3. Taylor expanded in a around 0 92.3%

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

      1. associate-*r/ 92.3%

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

      2. cancel-sign-sub-inv 92.3%

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

      3. metadata-eval 92.3%

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

      4. *-commutative 92.3%

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

      5. *-commutative 92.3%

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

      6. associate-*r* 92.3%

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

      7. fma-define 92.9%

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

      8. associate-*l/ 92.8%

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

      9. *-commutative 92.8%

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

      10. fma-define 92.2%

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

      11. +-commutative 92.2%

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

      12. fma-define 92.8%

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

   5. Simplified 92.8%

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

   6. Taylor expanded in z around 0 73.8%

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

   if -9.9999999999999995e-127 < (*.f64 x y) < 3.99999999999999996e-95

   1. Initial program 85.3%

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

   3. Taylor expanded in t around inf 87.9%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]

   4. Taylor expanded in t around inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r* 75.4%

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

      4. associate-*r/ 79.4%

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

      5. associate-/l* 79.4%

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

   6. Simplified 79.4%

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

   if 1.9999999999999999e-48 < (*.f64 x y) < 5.00000000000000018e-11

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 82.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-48} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{2 \cdot a}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x y) (* 2.0 a))))
   (if (<= (* x y) -1e-126)
     t_1
     (if (<= (* x y) 4e-95)
       (* t (* z (/ -4.5 a)))
       (if (<= (* x y) 2e-48)
         (* (* x y) (/ 0.5 a))
         (if (<= (* x y) 5e-11) (* -4.5 (/ (* z t) a)) t_1))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / (2.0 * a);
	double tmp;
	if ((x * y) <= -1e-126) {
		tmp = t_1;
	} else if ((x * y) <= 4e-95) {
		tmp = t * (z * (-4.5 / a));
	} else if ((x * y) <= 2e-48) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 5e-11) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) / (2.0d0 * a)
    if ((x * y) <= (-1d-126)) then
        tmp = t_1
    else if ((x * y) <= 4d-95) then
        tmp = t * (z * ((-4.5d0) / a))
    else if ((x * y) <= 2d-48) then
        tmp = (x * y) * (0.5d0 / a)
    else if ((x * y) <= 5d-11) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / (2.0 * a);
	double tmp;
	if ((x * y) <= -1e-126) {
		tmp = t_1;
	} else if ((x * y) <= 4e-95) {
		tmp = t * (z * (-4.5 / a));
	} else if ((x * y) <= 2e-48) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 5e-11) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) / (2.0 * a)
	tmp = 0
	if (x * y) <= -1e-126:
		tmp = t_1
	elif (x * y) <= 4e-95:
		tmp = t * (z * (-4.5 / a))
	elif (x * y) <= 2e-48:
		tmp = (x * y) * (0.5 / a)
	elif (x * y) <= 5e-11:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / Float64(2.0 * a))
	tmp = 0.0
	if (Float64(x * y) <= -1e-126)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-95)
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	elseif (Float64(x * y) <= 2e-48)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	elseif (Float64(x * y) <= 5e-11)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) / (2.0 * a);
	tmp = 0.0;
	if ((x * y) <= -1e-126)
		tmp = t_1;
	elseif ((x * y) <= 4e-95)
		tmp = t * (z * (-4.5 / a));
	elseif ((x * y) <= 2e-48)
		tmp = (x * y) * (0.5 / a);
	elseif ((x * y) <= 5e-11)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e-126], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-95], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-48], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-11], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -9.9999999999999995e-127 or 5.00000000000000018e-11 < (*.f64 x y)

   1. Initial program 91.9%

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

   3. Taylor expanded in x around inf 72.5%

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

   if -9.9999999999999995e-127 < (*.f64 x y) < 3.99999999999999996e-95

   1. Initial program 85.3%

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

   3. Taylor expanded in t around inf 87.9%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]

   4. Taylor expanded in t around inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r* 75.4%

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

      4. associate-*r/ 79.4%

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

      5. associate-/l* 79.4%

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

   6. Simplified 79.4%

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

   if 3.99999999999999996e-95 < (*.f64 x y) < 1.9999999999999999e-48

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. fma-define 99.6%

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

      8. associate-*l/ 100.0%

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

      9. *-commutative 100.0%

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

      10. fma-define 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 100.0%

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

   if 1.9999999999999999e-48 < (*.f64 x y) < 5.00000000000000018e-11

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 82.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 5e+277) (/ (- (* x y) t_1) (* 2.0 a)) (* -4.5 (* z (/ t a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= 5e+277) {
		tmp = ((x * y) - t_1) / (2.0 * a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= 5d+277) then
        tmp = ((x * y) - t_1) / (2.0d0 * a)
    else
        tmp = (-4.5d0) * (z * (t / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= 5e+277) {
		tmp = ((x * y) - t_1) / (2.0 * a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= 5e+277:
		tmp = ((x * y) - t_1) / (2.0 * a)
	else:
		tmp = -4.5 * (z * (t / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= 5e+277)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(2.0 * a));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= 5e+277)
		tmp = ((x * y) - t_1) / (2.0 * a);
	else
		tmp = -4.5 * (z * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+277], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999982e277

   1. Initial program 94.0%

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

   if 4.99999999999999982e277 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 59.6%

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

   3. Taylor expanded in y around inf 59.0%

      \[\leadsto \color{blue}{y \cdot \left(-4.5 \cdot \frac{t \cdot z}{a \cdot y} + 0.5 \cdot \frac{x}{a}\right)} \]

   4. Taylor expanded in y around 0 67.1%

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

      1. associate-*l/ 96.0%

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

      2. *-commutative 96.0%

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

   6. Simplified 96.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-25} \lor \neg \left(t \leq 8.6 \cdot 10^{+142}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.8e-25) (not (<= t 8.6e+142)))
   (* t (* z (/ -4.5 a)))
   (* x (* y (/ 0.5 a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.8e-25) || !(t <= 8.6e+142)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = x * (y * (0.5 / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-2.8d-25)) .or. (.not. (t <= 8.6d+142))) then
        tmp = t * (z * ((-4.5d0) / a))
    else
        tmp = x * (y * (0.5d0 / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.8e-25) || !(t <= 8.6e+142)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = x * (y * (0.5 / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.8e-25) or not (t <= 8.6e+142):
		tmp = t * (z * (-4.5 / a))
	else:
		tmp = x * (y * (0.5 / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.8e-25) || !(t <= 8.6e+142))
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.8e-25) || ~((t <= 8.6e+142)))
		tmp = t * (z * (-4.5 / a));
	else
		tmp = x * (y * (0.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.8e-25], N[Not[LessEqual[t, 8.6e+142]], $MachinePrecision]], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999988e-25 or 8.60000000000000025e142 < t

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 85.6%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]

   4. Taylor expanded in t around inf 58.5%

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

      1. associate-*r/ 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-*r* 58.4%

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

      4. associate-*r/ 68.3%

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

      5. associate-/l* 68.3%

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

   6. Simplified 68.3%

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

   if -2.79999999999999988e-25 < t < 8.60000000000000025e142

   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 67.9%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r* 67.9%

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

      4. associate-*r/ 65.5%

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

      5. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-25} \lor \neg \left(t \leq 8.6 \cdot 10^{+142}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-25} \lor \neg \left(t \leq 8.6 \cdot 10^{+142}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.3e-25) (not (<= t 8.6e+142)))
   (* t (* z (/ -4.5 a)))
   (* 0.5 (* x (/ y a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.3e-25) || !(t <= 8.6e+142)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-5.3d-25)) .or. (.not. (t <= 8.6d+142))) then
        tmp = t * (z * ((-4.5d0) / a))
    else
        tmp = 0.5d0 * (x * (y / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.3e-25) || !(t <= 8.6e+142)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (t <= -5.3e-25) or not (t <= 8.6e+142):
		tmp = t * (z * (-4.5 / a))
	else:
		tmp = 0.5 * (x * (y / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.3e-25) || !(t <= 8.6e+142))
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -5.3e-25) || ~((t <= 8.6e+142)))
		tmp = t * (z * (-4.5 / a));
	else
		tmp = 0.5 * (x * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.3e-25], N[Not[LessEqual[t, 8.6e+142]], $MachinePrecision]], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2999999999999997e-25 or 8.60000000000000025e142 < t

   1. Initial program 83.8%

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

   3. Taylor expanded in t around inf 85.6%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]

   4. Taylor expanded in t around inf 58.5%

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

      1. associate-*r/ 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-*r* 58.4%

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

      4. associate-*r/ 68.3%

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

      5. associate-/l* 68.3%

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

   6. Simplified 68.3%

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

   if -5.2999999999999997e-25 < t < 8.60000000000000025e142

   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 67.9%

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

      1. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-25} \lor \neg \left(t \leq 8.6 \cdot 10^{+142}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.85e-25)
   (* z (/ (* t -4.5) a))
   (if (<= t 1.6e+143) (* x (* y (/ 0.5 a))) (* t (* z (/ -4.5 a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.85e-25) {
		tmp = z * ((t * -4.5) / a);
	} else if (t <= 1.6e+143) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-3.85d-25)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if (t <= 1.6d+143) then
        tmp = x * (y * (0.5d0 / a))
    else
        tmp = t * (z * ((-4.5d0) / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.85e-25) {
		tmp = z * ((t * -4.5) / a);
	} else if (t <= 1.6e+143) {
		tmp = x * (y * (0.5 / a));
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.85e-25:
		tmp = z * ((t * -4.5) / a)
	elif t <= 1.6e+143:
		tmp = x * (y * (0.5 / a))
	else:
		tmp = t * (z * (-4.5 / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.85e-25)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif (t <= 1.6e+143)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	else
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.85e-25)
		tmp = z * ((t * -4.5) / a);
	elseif (t <= 1.6e+143)
		tmp = x * (y * (0.5 / a));
	else
		tmp = t * (z * (-4.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.85e-25], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+143], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8500000000000001e-25

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0 55.6%

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

      1. associate-*r/ 55.6%

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

      2. associate-*r* 55.5%

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

      3. *-commutative 55.5%

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

      4. associate-*r/ 63.0%

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

   5. Simplified 63.0%

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

   if -3.8500000000000001e-25 < t < 1.60000000000000008e143

   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 67.9%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r* 67.9%

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

      4. associate-*r/ 65.5%

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

      5. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

   if 1.60000000000000008e143 < t

   1. Initial program 76.2%

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

   3. Taylor expanded in t around inf 90.8%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]

   4. Taylor expanded in t around inf 64.1%

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

      1. associate-*r/ 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

      4. associate-*r/ 78.6%

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

      5. associate-/l* 78.7%

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

   6. Simplified 78.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{-25}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.3e-25)
   (* -4.5 (* z (/ t a)))
   (if (<= t 8.6e+142) (* 0.5 (* x (/ y a))) (* -4.5 (* t (/ z a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.3e-25) {
		tmp = -4.5 * (z * (t / a));
	} else if (t <= 8.6e+142) {
		tmp = 0.5 * (x * (y / a));
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-3.3d-25)) then
        tmp = (-4.5d0) * (z * (t / a))
    else if (t <= 8.6d+142) then
        tmp = 0.5d0 * (x * (y / a))
    else
        tmp = (-4.5d0) * (t * (z / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.3e-25) {
		tmp = -4.5 * (z * (t / a));
	} else if (t <= 8.6e+142) {
		tmp = 0.5 * (x * (y / a));
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.3e-25:
		tmp = -4.5 * (z * (t / a))
	elif t <= 8.6e+142:
		tmp = 0.5 * (x * (y / a))
	else:
		tmp = -4.5 * (t * (z / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.3e-25)
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	elseif (t <= 8.6e+142)
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.3e-25)
		tmp = -4.5 * (z * (t / a));
	elseif (t <= 8.6e+142)
		tmp = 0.5 * (x * (y / a));
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.3e-25], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e+142], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.2999999999999998e-25

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf 73.9%

      \[\leadsto \color{blue}{y \cdot \left(-4.5 \cdot \frac{t \cdot z}{a \cdot y} + 0.5 \cdot \frac{x}{a}\right)} \]

   4. Taylor expanded in y around 0 55.6%

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

      1. associate-*l/ 62.9%

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

      2. *-commutative 62.9%

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

   6. Simplified 62.9%

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

   if -3.2999999999999998e-25 < t < 8.60000000000000025e142

   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 67.9%

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

      1. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

   if 8.60000000000000025e142 < t

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0 64.1%

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

      1. associate-/l* 78.8%

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

   5. Simplified 78.8%

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

Alternative 11: 52.2% accurate, 1.9× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* -4.5 (* t (/ z a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return -4.5 * (t * (z / a));
}

Fortran

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

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return -4.5 * (t * (z / a));
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return -4.5 * (t * (z / a))

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(-4.5 * Float64(t * Float64(z / a)))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = -4.5 * (t * (z / a));
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in x around 0 44.2%

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

   1. associate-/l* 46.9%

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

5. Simplified 46.9%

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

Developer target: 93.7% 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 2024111 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

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

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