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

Percentage Accurate: 90.7% → 95.2%

Time: 9.7s

Alternatives: 11

Speedup: 0.8×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% 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: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot a\_m \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, 0.5 \cdot y, \left(\frac{z}{a\_m} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* 2.0 a_m) 5e-49)
    (/ (fma (* z -9.0) t (* x y)) (* 2.0 a_m))
    (fma (/ x a_m) (* 0.5 y) (* (* (/ z a_m) 4.5) (- t))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((2.0 * a_m) <= 5e-49) {
		tmp = fma((z * -9.0), t, (x * y)) / (2.0 * a_m);
	} else {
		tmp = fma((x / a_m), (0.5 * y), (((z / a_m) * 4.5) * -t));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(2.0 * a_m) <= 5e-49)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = fma(Float64(x / a_m), Float64(0.5 * y), Float64(Float64(Float64(z / a_m) * 4.5) * Float64(-t)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(2.0 * a$95$m), $MachinePrecision], 5e-49], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * N[(0.5 * y), $MachinePrecision] + N[(N[(N[(z / a$95$m), $MachinePrecision] * 4.5), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.9999999999999999e-49

   1. Initial program 92.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 92.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   if 4.9999999999999999e-49 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 86.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, y \cdot 0.5, \left(-t\right) \cdot \left(4.5 \cdot \frac{z}{a}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, 0.5 \cdot y, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{-4.5}{a\_m} \cdot z\right) \cdot t\\ t_2 := \frac{x \cdot y - \left(9 \cdot z\right) \cdot t}{2 \cdot a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{-4.5}{a\_m} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ -4.5 a_m) z) t))
        (t_2 (/ (- (* x y) (* (* 9.0 z) t)) (* 2.0 a_m))))
   (*
    a_s
    (if (<= t_2 -2e+307)
      t_1
      (if (<= t_2 5e+273) (* (/ -4.5 a_m) (* t z)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((-4.5 / a_m) * z) * t;
	double t_2 = ((x * y) - ((9.0 * z) * t)) / (2.0 * a_m);
	double tmp;
	if (t_2 <= -2e+307) {
		tmp = t_1;
	} else if (t_2 <= 5e+273) {
		tmp = (-4.5 / a_m) * (t * z);
	} 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.
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) / a_m) * z) * t
    t_2 = ((x * y) - ((9.0d0 * z) * t)) / (2.0d0 * a_m)
    if (t_2 <= (-2d+307)) then
        tmp = t_1
    else if (t_2 <= 5d+273) then
        tmp = ((-4.5d0) / a_m) * (t * z)
    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;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((-4.5 / a_m) * z) * t;
	double t_2 = ((x * y) - ((9.0 * z) * t)) / (2.0 * a_m);
	double tmp;
	if (t_2 <= -2e+307) {
		tmp = t_1;
	} else if (t_2 <= 5e+273) {
		tmp = (-4.5 / a_m) * (t * z);
	} 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])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((-4.5 / a_m) * z) * t
	t_2 = ((x * y) - ((9.0 * z) * t)) / (2.0 * a_m)
	tmp = 0
	if t_2 <= -2e+307:
		tmp = t_1
	elif t_2 <= 5e+273:
		tmp = (-4.5 / a_m) * (t * z)
	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])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(-4.5 / a_m) * z) * t)
	t_2 = Float64(Float64(Float64(x * y) - Float64(Float64(9.0 * z) * t)) / Float64(2.0 * a_m))
	tmp = 0.0
	if (t_2 <= -2e+307)
		tmp = t_1;
	elseif (t_2 <= 5e+273)
		tmp = Float64(Float64(-4.5 / a_m) * Float64(t * z));
	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])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((-4.5 / a_m) * z) * t;
	t_2 = ((x * y) - ((9.0 * z) * t)) / (2.0 * a_m);
	tmp = 0.0;
	if (t_2 <= -2e+307)
		tmp = t_1;
	elseif (t_2 <= 5e+273)
		tmp = (-4.5 / a_m) * (t * z);
	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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(-4.5 / a$95$m), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -2e+307], t$95$1, If[LessEqual[t$95$2, 5e+273], N[(N[(-4.5 / a$95$m), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -1.99999999999999997e307 or 4.99999999999999961e273 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 76.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.6%

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

   if -1.99999999999999997e307 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 4.99999999999999961e273

   1. Initial program 99.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 55.1%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(9 \cdot z\right) \cdot t}{2 \cdot a} \leq -2 \cdot 10^{+307}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \mathbf{elif}\;\frac{x \cdot y - \left(9 \cdot z\right) \cdot t}{2 \cdot a} \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot a\_m \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a\_m}, -4.5 \cdot t, \left(0.5 \cdot \frac{x}{a\_m}\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* 2.0 a_m) 5e-49)
    (/ (fma (* z -9.0) t (* x y)) (* 2.0 a_m))
    (fma (/ z a_m) (* -4.5 t) (* (* 0.5 (/ x a_m)) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((2.0 * a_m) <= 5e-49) {
		tmp = fma((z * -9.0), t, (x * y)) / (2.0 * a_m);
	} else {
		tmp = fma((z / a_m), (-4.5 * t), ((0.5 * (x / a_m)) * y));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(2.0 * a_m) <= 5e-49)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = fma(Float64(z / a_m), Float64(-4.5 * t), Float64(Float64(0.5 * Float64(x / a_m)) * y));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(2.0 * a$95$m), $MachinePrecision], 5e-49], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z / a$95$m), $MachinePrecision] * N[(-4.5 * t), $MachinePrecision] + N[(N[(0.5 * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.9999999999999999e-49

   1. Initial program 92.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 92.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   if 4.9999999999999999e-49 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 86.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 85.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. sub-div N/A

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

      12. frac-2neg N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

   8. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{a}, -4.5 \cdot t, \left(\frac{x}{a} \cdot 0.5\right) \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, -4.5 \cdot t, \left(0.5 \cdot \frac{x}{a}\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot a\_m \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m} \cdot 0.5, x, \left(-4.5 \cdot \frac{z}{a\_m}\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* 2.0 a_m) 5e-49)
    (/ (fma (* z -9.0) t (* x y)) (* 2.0 a_m))
    (fma (* (/ y a_m) 0.5) x (* (* -4.5 (/ z a_m)) t)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((2.0 * a_m) <= 5e-49) {
		tmp = fma((z * -9.0), t, (x * y)) / (2.0 * a_m);
	} else {
		tmp = fma(((y / a_m) * 0.5), x, ((-4.5 * (z / a_m)) * t));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(2.0 * a_m) <= 5e-49)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = fma(Float64(Float64(y / a_m) * 0.5), x, Float64(Float64(-4.5 * Float64(z / a_m)) * t));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(2.0 * a$95$m), $MachinePrecision], 5e-49], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x + N[(N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.9999999999999999e-49

   1. Initial program 92.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 92.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   if 4.9999999999999999e-49 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 86.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 95.3

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

   8. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a} \cdot 0.5, x, \left(\frac{z}{a} \cdot -4.5\right) \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a} \cdot 0.5, x, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(t \cdot z\right) \cdot -4.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -1e-31)
    (* (* (/ y a_m) 0.5) x)
    (if (<= (* x y) 2e-42) (/ (* (* t z) -4.5) a_m) (* (* 0.5 y) (/ x a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = ((t * z) * -4.5) / a_m;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-1d-31)) then
        tmp = ((y / a_m) * 0.5d0) * x
    else if ((x * y) <= 2d-42) then
        tmp = ((t * z) * (-4.5d0)) / a_m
    else
        tmp = (0.5d0 * y) * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = ((t * z) * -4.5) / a_m;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -1e-31:
		tmp = ((y / a_m) * 0.5) * x
	elif (x * y) <= 2e-42:
		tmp = ((t * z) * -4.5) / a_m
	else:
		tmp = (0.5 * y) * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -1e-31)
		tmp = Float64(Float64(Float64(y / a_m) * 0.5) * x);
	elseif (Float64(x * y) <= 2e-42)
		tmp = Float64(Float64(Float64(t * z) * -4.5) / a_m);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -1e-31)
		tmp = ((y / a_m) * 0.5) * x;
	elseif ((x * y) <= 2e-42)
		tmp = ((t * z) * -4.5) / a_m;
	else
		tmp = (0.5 * y) * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-31], N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-42], N[(N[(N[(t * z), $MachinePrecision] * -4.5), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e-31

   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 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -1e-31 < (*.f64 x y) < 2.00000000000000008e-42

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 83.2%

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

   if 2.00000000000000008e-42 < (*.f64 x y)

   1. Initial program 87.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(t \cdot z\right) \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -1e-31)
    (* (* (/ y a_m) 0.5) x)
    (if (<= (* x y) 2e-42) (/ (* (* -4.5 t) z) a_m) (* (* 0.5 y) (/ x a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-1d-31)) then
        tmp = ((y / a_m) * 0.5d0) * x
    else if ((x * y) <= 2d-42) then
        tmp = (((-4.5d0) * t) * z) / a_m
    else
        tmp = (0.5d0 * y) * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -1e-31:
		tmp = ((y / a_m) * 0.5) * x
	elif (x * y) <= 2e-42:
		tmp = ((-4.5 * t) * z) / a_m
	else:
		tmp = (0.5 * y) * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -1e-31)
		tmp = Float64(Float64(Float64(y / a_m) * 0.5) * x);
	elseif (Float64(x * y) <= 2e-42)
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a_m);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -1e-31)
		tmp = ((y / a_m) * 0.5) * x;
	elseif ((x * y) <= 2e-42)
		tmp = ((-4.5 * t) * z) / a_m;
	else
		tmp = (0.5 * y) * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-31], N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-42], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e-31

   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 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -1e-31 < (*.f64 x y) < 2.00000000000000008e-42

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 83.0%

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

   if 2.00000000000000008e-42 < (*.f64 x y)

   1. Initial program 87.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-4.5}{a\_m} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -1e-31)
    (* (* (/ y a_m) 0.5) x)
    (if (<= (* x y) 2e-42) (* (/ -4.5 a_m) (* t z)) (* (* 0.5 y) (/ x a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = (-4.5 / a_m) * (t * z);
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-1d-31)) then
        tmp = ((y / a_m) * 0.5d0) * x
    else if ((x * y) <= 2d-42) then
        tmp = ((-4.5d0) / a_m) * (t * z)
    else
        tmp = (0.5d0 * y) * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = ((y / a_m) * 0.5) * x;
	} else if ((x * y) <= 2e-42) {
		tmp = (-4.5 / a_m) * (t * z);
	} else {
		tmp = (0.5 * y) * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -1e-31:
		tmp = ((y / a_m) * 0.5) * x
	elif (x * y) <= 2e-42:
		tmp = (-4.5 / a_m) * (t * z)
	else:
		tmp = (0.5 * y) * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -1e-31)
		tmp = Float64(Float64(Float64(y / a_m) * 0.5) * x);
	elseif (Float64(x * y) <= 2e-42)
		tmp = Float64(Float64(-4.5 / a_m) * Float64(t * z));
	else
		tmp = Float64(Float64(0.5 * y) * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -1e-31)
		tmp = ((y / a_m) * 0.5) * x;
	elseif ((x * y) <= 2e-42)
		tmp = (-4.5 / a_m) * (t * z);
	else
		tmp = (0.5 * y) * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-31], N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-42], N[(N[(-4.5 / a$95$m), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e-31

   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 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -1e-31 < (*.f64 x y) < 2.00000000000000008e-42

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 83.1%

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

   if 2.00000000000000008e-42 < (*.f64 x y)

   1. Initial program 87.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{y}{a\_m} \cdot 0.5\right) \cdot x\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-43}:\\ \;\;\;\;\frac{-4.5}{a\_m} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ y a_m) 0.5) x)))
   (*
    a_s
    (if (<= (* x y) -1e-31)
      t_1
      (if (<= (* x y) 1e-43) (* (/ -4.5 a_m) (* t z)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = t_1;
	} else if ((x * y) <= 1e-43) {
		tmp = (-4.5 / a_m) * (t * z);
	} 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.
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 = ((y / a_m) * 0.5d0) * x
    if ((x * y) <= (-1d-31)) then
        tmp = t_1
    else if ((x * y) <= 1d-43) then
        tmp = ((-4.5d0) / a_m) * (t * z)
    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;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((y / a_m) * 0.5) * x;
	double tmp;
	if ((x * y) <= -1e-31) {
		tmp = t_1;
	} else if ((x * y) <= 1e-43) {
		tmp = (-4.5 / a_m) * (t * z);
	} 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])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((y / a_m) * 0.5) * x
	tmp = 0
	if (x * y) <= -1e-31:
		tmp = t_1
	elif (x * y) <= 1e-43:
		tmp = (-4.5 / a_m) * (t * z)
	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])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(y / a_m) * 0.5) * x)
	tmp = 0.0
	if (Float64(x * y) <= -1e-31)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-43)
		tmp = Float64(Float64(-4.5 / a_m) * Float64(t * z));
	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])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((y / a_m) * 0.5) * x;
	tmp = 0.0;
	if ((x * y) <= -1e-31)
		tmp = t_1;
	elseif ((x * y) <= 1e-43)
		tmp = (-4.5 / a_m) * (t * z);
	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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(y / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-31], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-43], N[(N[(-4.5 / a$95$m), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e-31 or 1.00000000000000008e-43 < (*.f64 x y)

   1. Initial program 86.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -1e-31 < (*.f64 x y) < 1.00000000000000008e-43

   1. Initial program 96.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 83.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-43}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 55.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -inf.0 < (*.f64 x y)

   1. Initial program 93.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 93.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

4. Final simplification 93.4%

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

Alternative 10: 92.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 55.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -inf.0 < (*.f64 x y)

   1. Initial program 93.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval 93.7

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

   4. Applied rewrites 93.7%

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

4. Final simplification 93.7%

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

Alternative 11: 51.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 51.0

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

5. Applied rewrites 51.0%

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

7. Applied rewrites 51.8%

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

8. Final simplification 51.8%

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

Developer Target 1: 94.4% 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 2024276 
(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)))