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

Percentage Accurate: 91.5% → 94.6%

Time: 11.5s

Alternatives: 7

Speedup: 0.7×

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

Specification

Math

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

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: 91.5% 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: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \cdot 2 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a\_m}, z \cdot -4.5, x \cdot \frac{y}{a\_m \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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) 2e+47)
    (/ (fma (* z -9.0) t (* x y)) (* a_m 2.0))
    (fma (/ t a_m) (* z -4.5) (* x (/ y (* 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) <= 2e+47) {
		tmp = fma((z * -9.0), t, (x * y)) / (a_m * 2.0);
	} else {
		tmp = fma((t / a_m), (z * -4.5), (x * (y / (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) <= 2e+47)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a_m * 2.0));
	else
		tmp = fma(Float64(t / a_m), Float64(z * -4.5), Float64(x * Float64(y / 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], 2e+47], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t / a$95$m), $MachinePrecision] * N[(z * -4.5), $MachinePrecision] + N[(x * N[(y / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 2.0000000000000001e47

   1. Initial program 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 95.1

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

   4. Applied rewrites 95.1%

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

   if 2.0000000000000001e47 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 88.5

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

   6. Applied rewrites 88.5%

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

Alternative 2: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\frac{z}{a\_m} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;t\_1 \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 -5e+54)
      (* (/ z a_m) (* t -4.5))
      (if (<= t_1 400.0) (* 0.5 (* x (/ 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 t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+54) {
		tmp = (z / a_m) * (t * -4.5);
	} else if (t_1 <= 400.0) {
		tmp = 0.5 * (x * (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) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-5d+54)) then
        tmp = (z / a_m) * (t * (-4.5d0))
    else if (t_1 <= 400.0d0) then
        tmp = 0.5d0 * (x * (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 t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+54) {
		tmp = (z / a_m) * (t * -4.5);
	} else if (t_1 <= 400.0) {
		tmp = 0.5 * (x * (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):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -5e+54:
		tmp = (z / a_m) * (t * -4.5)
	elif t_1 <= 400.0:
		tmp = 0.5 * (x * (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)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+54)
		tmp = Float64(Float64(z / a_m) * Float64(t * -4.5));
	elseif (t_1 <= 400.0)
		tmp = Float64(0.5 * Float64(x * 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)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e+54)
		tmp = (z / a_m) * (t * -4.5);
	elseif (t_1 <= 400.0)
		tmp = 0.5 * (x * (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_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -5e+54], N[(N[(z / a$95$m), $MachinePrecision] * N[(t * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 400.0], N[(0.5 * N[(x * 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 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000005e54

   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 x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.9%

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

   if -5.00000000000000005e54 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 400

   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 x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.5%

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

   9. Applied rewrites 32.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 74.3

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

   12. Applied rewrites 74.3%

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

   if 400 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_1 -5e+54)
      (* -4.5 (* t (/ z a_m)))
      (if (<= t_1 400.0) (* 0.5 (* x (/ 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 t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+54) {
		tmp = -4.5 * (t * (z / a_m));
	} else if (t_1 <= 400.0) {
		tmp = 0.5 * (x * (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) :: t_1
    real(8) :: tmp
    t_1 = t * (z * 9.0d0)
    if (t_1 <= (-5d+54)) then
        tmp = (-4.5d0) * (t * (z / a_m))
    else if (t_1 <= 400.0d0) then
        tmp = 0.5d0 * (x * (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 t_1 = t * (z * 9.0);
	double tmp;
	if (t_1 <= -5e+54) {
		tmp = -4.5 * (t * (z / a_m));
	} else if (t_1 <= 400.0) {
		tmp = 0.5 * (x * (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):
	t_1 = t * (z * 9.0)
	tmp = 0
	if t_1 <= -5e+54:
		tmp = -4.5 * (t * (z / a_m))
	elif t_1 <= 400.0:
		tmp = 0.5 * (x * (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)
	t_1 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_1 <= -5e+54)
		tmp = Float64(-4.5 * Float64(t * Float64(z / a_m)));
	elseif (t_1 <= 400.0)
		tmp = Float64(0.5 * Float64(x * 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)
	t_1 = t * (z * 9.0);
	tmp = 0.0;
	if (t_1 <= -5e+54)
		tmp = -4.5 * (t * (z / a_m));
	elseif (t_1 <= 400.0)
		tmp = 0.5 * (x * (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_] := Block[{t$95$1 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -5e+54], N[(-4.5 * N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 400.0], N[(0.5 * N[(x * 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 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000005e54

   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 x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -5.00000000000000005e54 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 400

   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 x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.5%

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

   9. Applied rewrites 32.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 74.3

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

   12. Applied rewrites 74.3%

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

   if 400 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := -4.5 \cdot \left(t \cdot \frac{z}{a\_m}\right)\\ t_2 := t \cdot \left(z \cdot 9\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (* -4.5 (* t (/ z a_m)))) (t_2 (* t (* z 9.0))))
   (*
    a_s
    (if (<= t_2 -5e+54) t_1 (if (<= t_2 400.0) (* 0.5 (* x (/ y a_m))) 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 = -4.5 * (t * (z / a_m));
	double t_2 = t * (z * 9.0);
	double tmp;
	if (t_2 <= -5e+54) {
		tmp = t_1;
	} else if (t_2 <= 400.0) {
		tmp = 0.5 * (x * (y / a_m));
	} 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 = (-4.5d0) * (t * (z / a_m))
    t_2 = t * (z * 9.0d0)
    if (t_2 <= (-5d+54)) then
        tmp = t_1
    else if (t_2 <= 400.0d0) then
        tmp = 0.5d0 * (x * (y / a_m))
    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 = -4.5 * (t * (z / a_m));
	double t_2 = t * (z * 9.0);
	double tmp;
	if (t_2 <= -5e+54) {
		tmp = t_1;
	} else if (t_2 <= 400.0) {
		tmp = 0.5 * (x * (y / a_m));
	} 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 = -4.5 * (t * (z / a_m))
	t_2 = t * (z * 9.0)
	tmp = 0
	if t_2 <= -5e+54:
		tmp = t_1
	elif t_2 <= 400.0:
		tmp = 0.5 * (x * (y / a_m))
	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(-4.5 * Float64(t * Float64(z / a_m)))
	t_2 = Float64(t * Float64(z * 9.0))
	tmp = 0.0
	if (t_2 <= -5e+54)
		tmp = t_1;
	elseif (t_2 <= 400.0)
		tmp = Float64(0.5 * Float64(x * Float64(y / a_m)));
	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 = -4.5 * (t * (z / a_m));
	t_2 = t * (z * 9.0);
	tmp = 0.0;
	if (t_2 <= -5e+54)
		tmp = t_1;
	elseif (t_2 <= 400.0)
		tmp = 0.5 * (x * (y / a_m));
	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[(-4.5 * N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -5e+54], t$95$1, If[LessEqual[t$95$2, 400.0], N[(0.5 * N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.00000000000000005e54 or 400 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 91.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -5.00000000000000005e54 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 400

   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 x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.5%

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

   9. Applied rewrites 32.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 74.3

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

   12. Applied rewrites 74.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t \cdot \left(z \cdot 9\right) \leq 400:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a\_m} \cdot \left(z \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (<= (* t (* z 9.0)) 2e+297)
    (/ (fma (* z -9.0) t (* x y)) (* a_m 2.0))
    (* (/ t a_m) (* z -4.5)))))

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 ((t * (z * 9.0)) <= 2e+297) {
		tmp = fma((z * -9.0), t, (x * y)) / (a_m * 2.0);
	} else {
		tmp = (t / a_m) * (z * -4.5);
	}
	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(t * Float64(z * 9.0)) <= 2e+297)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a_m * 2.0));
	else
		tmp = Float64(Float64(t / a_m) * Float64(z * -4.5));
	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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t / a$95$m), $MachinePrecision] * N[(z * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e297

   1. Initial program 94.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 94.3

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

   4. Applied rewrites 94.3%

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

   if 2e297 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot \left(z \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, -9 \cdot t, x \cdot y\right) \cdot \frac{0.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a\_m} \cdot \left(z \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (<= (* t (* z 9.0)) 2e+297)
    (* (fma z (* -9.0 t) (* x y)) (/ 0.5 a_m))
    (* (/ t a_m) (* z -4.5)))))

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 ((t * (z * 9.0)) <= 2e+297) {
		tmp = fma(z, (-9.0 * t), (x * y)) * (0.5 / a_m);
	} else {
		tmp = (t / a_m) * (z * -4.5);
	}
	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(t * Float64(z * 9.0)) <= 2e+297)
		tmp = Float64(fma(z, Float64(-9.0 * t), Float64(x * y)) * Float64(0.5 / a_m));
	else
		tmp = Float64(Float64(t / a_m) * Float64(z * -4.5));
	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[(t * N[(z * 9.0), $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(z * N[(-9.0 * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t / a$95$m), $MachinePrecision] * N[(z * -4.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e297

   1. Initial program 94.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval 94.2

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

   4. Applied rewrites 94.2%

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

   if 2e297 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(z \cdot 9\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(z, -9 \cdot t, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot \left(z \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 1.6× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 92.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 54.0

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

5. Applied rewrites 54.0%

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

Developer Target 1: 93.9% accurate, 0.6× 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 2024232 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

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