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

Percentage Accurate: 91.2% → 95.1%

Time: 8.9s

Alternatives: 13

Speedup: 0.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% 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: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot a\_m \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, 0.5 \cdot x, \left(\frac{z}{a\_m} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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.
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) 2e+45)
    (/ (fma (* t -9.0) z (* x y)) (* 2.0 a_m))
    (fma (/ y a_m) (* 0.5 x) (* (* (/ 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);
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) <= 2e+45) {
		tmp = fma((t * -9.0), z, (x * y)) / (2.0 * a_m);
	} else {
		tmp = fma((y / a_m), (0.5 * x), (((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])
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) <= 2e+45)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = fma(Float64(y / a_m), Float64(0.5 * x), 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.
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], 2e+45], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * N[(0.5 * x), $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)) < 1.9999999999999999e45

   1. Initial program 94.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 94.5

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

      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 94.5

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

   4. Applied rewrites 94.5%

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

   if 1.9999999999999999e45 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 84.7%

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

      1. 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. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. times-frac N/A

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

      25. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, \left(\frac{z}{a} \cdot 4.5\right) \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a\_m}, -4.5 \cdot z, \left(\frac{0.5}{a\_m} \cdot x\right) \cdot y\right)\\ \end{array} \end{array} \]

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.
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 (<= (* (* 9.0 z) t) 2e+87)
    (/ (fma (* t -9.0) z (* x y)) (* 2.0 a_m))
    (fma (/ t a_m) (* -4.5 z) (* (* (/ 0.5 a_m) x) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((9.0 * z) * t) <= 2e+87) {
		tmp = fma((t * -9.0), z, (x * y)) / (2.0 * a_m);
	} else {
		tmp = fma((t / a_m), (-4.5 * z), (((0.5 / a_m) * 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])
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(Float64(9.0 * z) * t) <= 2e+87)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = fma(Float64(t / a_m), Float64(-4.5 * z), Float64(Float64(Float64(0.5 / a_m) * 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.
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[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision], 2e+87], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t / a$95$m), $MachinePrecision] * N[(-4.5 * z), $MachinePrecision] + N[(N[(N[(0.5 / a$95$m), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.9999999999999999e87

   1. Initial program 95.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 95.8

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

      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 95.8

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

   4. Applied rewrites 95.8%

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

   if 1.9999999999999999e87 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. metadata-eval 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, -4.5 \cdot z, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{x}{a\_m} \cdot 0.5\right) \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{\frac{a\_m}{x}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -5e+41)
    (* (* (/ x a_m) 0.5) y)
    (if (<= (* x y) 2e+31) (* (/ (* -4.5 z) a_m) t) (/ (* 0.5 y) (/ a_m x))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = ((x / a_m) * 0.5) * y;
	} else if ((x * y) <= 2e+31) {
		tmp = ((-4.5 * z) / a_m) * t;
	} else {
		tmp = (0.5 * y) / (a_m / x);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-5d+41)) then
        tmp = ((x / a_m) * 0.5d0) * y
    else if ((x * y) <= 2d+31) then
        tmp = (((-4.5d0) * z) / a_m) * t
    else
        tmp = (0.5d0 * y) / (a_m / x)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = ((x / a_m) * 0.5) * y;
	} else if ((x * y) <= 2e+31) {
		tmp = ((-4.5 * z) / a_m) * t;
	} else {
		tmp = (0.5 * y) / (a_m / x);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = ((x / a_m) * 0.5) * y
	elif (x * y) <= 2e+31:
		tmp = ((-4.5 * z) / a_m) * t
	else:
		tmp = (0.5 * y) / (a_m / x)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = Float64(Float64(Float64(x / a_m) * 0.5) * y);
	elseif (Float64(x * y) <= 2e+31)
		tmp = Float64(Float64(Float64(-4.5 * z) / a_m) * t);
	else
		tmp = Float64(Float64(0.5 * y) / Float64(a_m / x));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = ((x / a_m) * 0.5) * y;
	elseif ((x * y) <= 2e+31)
		tmp = ((-4.5 * z) / a_m) * t;
	else
		tmp = (0.5 * y) / (a_m / x);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], N[(N[(N[(x / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+31], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / N[(a$95$m / x), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5.00000000000000022e41

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 1.9999999999999999e31

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.4%

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

   if 1.9999999999999999e31 < (*.f64 x y)

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 81.8%

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

Alternative 4: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* (* 9.0 z) t) 1e+259)
    (/ (fma (* t -9.0) z (* x y)) (* 2.0 a_m))
    (* (/ (* -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);
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 (((9.0 * z) * t) <= 1e+259) {
		tmp = fma((t * -9.0), z, (x * y)) / (2.0 * a_m);
	} else {
		tmp = ((-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])
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(Float64(9.0 * z) * t) <= 1e+259)
		tmp = Float64(fma(Float64(t * -9.0), z, Float64(x * y)) / Float64(2.0 * a_m));
	else
		tmp = Float64(Float64(Float64(-4.5 * 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.
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[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision], 1e+259], N[(N[(N[(t * -9.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.999999999999999e258

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 95.7

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

      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 95.7

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

   4. Applied rewrites 95.7%

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

   if 9.999999999999999e258 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 52.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.9%

      \[\leadsto \frac{-4.5 \cdot z}{a} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t \cdot -9, z, x \cdot y\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot -9\right)}{2 \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* (* 9.0 z) t) 1e+259)
    (/ (fma y x (* (* z t) -9.0)) (* 2.0 a_m))
    (* (/ (* -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);
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 (((9.0 * z) * t) <= 1e+259) {
		tmp = fma(y, x, ((z * t) * -9.0)) / (2.0 * a_m);
	} else {
		tmp = ((-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])
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(Float64(9.0 * z) * t) <= 1e+259)
		tmp = Float64(fma(y, x, Float64(Float64(z * t) * -9.0)) / Float64(2.0 * a_m));
	else
		tmp = Float64(Float64(Float64(-4.5 * 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.
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[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision], 1e+259], N[(N[(y * x + N[(N[(z * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.999999999999999e258

   1. Initial program 95.7%

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 95.7

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

   4. Applied rewrites 95.7%

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

   if 9.999999999999999e258 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 52.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.9%

      \[\leadsto \frac{-4.5 \cdot z}{a} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot -9\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right) \cdot \frac{0.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* (* 9.0 z) t) 1e+259)
    (* (fma (* z t) -9.0 (* x y)) (/ 0.5 a_m))
    (* (/ (* -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);
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 (((9.0 * z) * t) <= 1e+259) {
		tmp = fma((z * t), -9.0, (x * y)) * (0.5 / a_m);
	} else {
		tmp = ((-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])
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(Float64(9.0 * z) * t) <= 1e+259)
		tmp = Float64(fma(Float64(z * t), -9.0, Float64(x * y)) * Float64(0.5 / a_m));
	else
		tmp = Float64(Float64(Float64(-4.5 * 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.
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[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision], 1e+259], N[(N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 9.999999999999999e258

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 95.6

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

   4. Applied rewrites 95.6%

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

   if 9.999999999999999e258 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 52.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.9%

      \[\leadsto \frac{-4.5 \cdot z}{a} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t, -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{x}{a\_m} \cdot 0.5\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ x a_m) 0.5) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 2e+31) (* (/ (* -4.5 z) a_m) t) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = ((-4.5 * z) / a_m) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x / a_m) * 0.5d0) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 2d+31) then
        tmp = (((-4.5d0) * z) / a_m) * t
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = ((-4.5 * z) / a_m) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x / a_m) * 0.5) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 2e+31:
		tmp = ((-4.5 * z) / a_m) * t
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x / a_m) * 0.5) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+31)
		tmp = Float64(Float64(Float64(-4.5 * z) / a_m) * t);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x / a_m) * 0.5) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 2e+31)
		tmp = ((-4.5 * z) / a_m) * t;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+31], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 1.9999999999999999e31 < (*.f64 x y)

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 1.9999999999999999e31

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.4%

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

Alternative 8: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{x}{a\_m} \cdot 0.5\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ x a_m) 0.5) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 2e+31) (* (* -4.5 t) (/ z a_m)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = (-4.5 * t) * (z / a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x / a_m) * 0.5d0) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 2d+31) then
        tmp = ((-4.5d0) * t) * (z / a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = (-4.5 * t) * (z / a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x / a_m) * 0.5) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 2e+31:
		tmp = (-4.5 * t) * (z / a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x / a_m) * 0.5) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+31)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x / a_m) * 0.5) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 2e+31)
		tmp = (-4.5 * t) * (z / a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+31], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 1.9999999999999999e31 < (*.f64 x y)

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 1.9999999999999999e31

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{x}{a\_m} \cdot 0.5\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(-4.5 \cdot \frac{z}{a\_m}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ x a_m) 0.5) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 2e+31) (* (* -4.5 (/ z a_m)) t) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = (-4.5 * (z / a_m)) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x / a_m) * 0.5d0) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 2d+31) then
        tmp = ((-4.5d0) * (z / a_m)) * t
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 2e+31) {
		tmp = (-4.5 * (z / a_m)) * t;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x / a_m) * 0.5) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 2e+31:
		tmp = (-4.5 * (z / a_m)) * t
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x / a_m) * 0.5) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+31)
		tmp = Float64(Float64(-4.5 * Float64(z / a_m)) * t);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x / a_m) * 0.5) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 2e+31)
		tmp = (-4.5 * (z / a_m)) * t;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+31], N[(N[(-4.5 * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 1.9999999999999999e31 < (*.f64 x y)

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 1.9999999999999999e31

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\left(-4.5 \cdot \frac{z}{a}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{x}{a\_m} \cdot 0.5\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{t}{a\_m} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ x a_m) 0.5) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 5e+16) (* (* (/ t a_m) z) -4.5) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = ((t / a_m) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x / a_m) * 0.5d0) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 5d+16) then
        tmp = ((t / a_m) * z) * (-4.5d0)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x / a_m) * 0.5) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = ((t / a_m) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x / a_m) * 0.5) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 5e+16:
		tmp = ((t / a_m) * z) * -4.5
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x / a_m) * 0.5) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+16)
		tmp = Float64(Float64(Float64(t / a_m) * z) * -4.5);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x / a_m) * 0.5) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 5e+16)
		tmp = ((t / a_m) * z) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x / a$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+16], N[(N[(N[(t / a$95$m), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 5e16 < (*.f64 x y)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 5e16

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 76.4%

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

Alternative 11: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a\_m} \cdot x\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{t}{a\_m} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ 0.5 a_m) x) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 5e+16) (* (* (/ t a_m) z) -4.5) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 / a_m) * x) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = ((t / a_m) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.5d0 / a_m) * x) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 5d+16) then
        tmp = ((t / a_m) * z) * (-4.5d0)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 / a_m) * x) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = ((t / a_m) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((0.5 / a_m) * x) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 5e+16:
		tmp = ((t / a_m) * z) * -4.5
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(0.5 / a_m) * x) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+16)
		tmp = Float64(Float64(Float64(t / a_m) * z) * -4.5);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((0.5 / a_m) * x) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 5e+16)
		tmp = ((t / a_m) * z) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(0.5 / a$95$m), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+16], N[(N[(N[(t / a$95$m), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 5e16 < (*.f64 x y)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 5e16

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 76.4%

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

Alternative 12: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a\_m} \cdot x\right) \cdot y\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(-4.5 \cdot \frac{t}{a\_m}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ 0.5 a_m) x) y)))
   (*
    a_s
    (if (<= (* x y) -5e+41)
      t_1
      (if (<= (* x y) 5e+16) (* (* -4.5 (/ t a_m)) z) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 / a_m) * x) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = (-4.5 * (t / a_m)) * 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.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((0.5d0 / a_m) * x) * y
    if ((x * y) <= (-5d+41)) then
        tmp = t_1
    else if ((x * y) <= 5d+16) then
        tmp = ((-4.5d0) * (t / a_m)) * 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;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((0.5 / a_m) * x) * y;
	double tmp;
	if ((x * y) <= -5e+41) {
		tmp = t_1;
	} else if ((x * y) <= 5e+16) {
		tmp = (-4.5 * (t / a_m)) * 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])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((0.5 / a_m) * x) * y
	tmp = 0
	if (x * y) <= -5e+41:
		tmp = t_1
	elif (x * y) <= 5e+16:
		tmp = (-4.5 * (t / a_m)) * 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])
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(0.5 / a_m) * x) * y)
	tmp = 0.0
	if (Float64(x * y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+16)
		tmp = Float64(Float64(-4.5 * Float64(t / a_m)) * 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])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((0.5 / a_m) * x) * y;
	tmp = 0.0;
	if ((x * y) <= -5e+41)
		tmp = t_1;
	elseif ((x * y) <= 5e+16)
		tmp = (-4.5 * (t / a_m)) * 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.
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[(0.5 / a$95$m), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+16], N[(N[(-4.5 * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000022e41 or 5e16 < (*.f64 x y)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.7%

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

   if -5.00000000000000022e41 < (*.f64 x y) < 5e16

   1. Initial program 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 92.9

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

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

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 92.9

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 76.4

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

   7. Applied rewrites 76.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot x\right) \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(-4.5 \cdot \frac{t}{a}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/l/ N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 92.1

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

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

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval 92.1

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 92.1

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

4. Applied rewrites 92.1%

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

5. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 49.0

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

7. Applied rewrites 49.0%

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

8. Final simplification 49.0%

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

Developer Target 1: 94.6% 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 2024295 
(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)))