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

Percentage Accurate: 91.4% → 94.4%

Time: 8.6s

Alternatives: 9

Speedup: 1.1×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+255}:\\ \;\;\;\;\mathsf{fma}\left(-z, 4.5 \cdot \frac{t}{a}, \left(x \cdot \frac{0.5}{a}\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) -4e+255)
   (fma (- z) (* 4.5 (/ t a)) (* (* x (/ 0.5 a)) y))
   (/ (fma (* -9.0 z) t (* y x)) (* a 2.0))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= -4e+255) {
		tmp = fma(-z, (4.5 * (t / a)), ((x * (0.5 / a)) * y));
	} else {
		tmp = fma((-9.0 * z), t, (y * x)) / (a * 2.0);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= -4e+255)
		tmp = fma(Float64(-z), Float64(4.5 * Float64(t / a)), Float64(Float64(x * Float64(0.5 / a)) * y));
	else
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], -4e+255], N[((-z) * N[(4.5 * N[(t / a), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -3.99999999999999995e255

   1. Initial program 84.0%

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

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

      8. associate-*l* N/A

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

      9. associate-/l* N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if -3.99999999999999995e255 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 96.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 96.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 2: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+240}\right):\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+240)))
     (* (* (/ t a) -4.5) z)
     (* (* t z) (/ -4.5 a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+240)) {
		tmp = ((t / a) * -4.5) * z;
	} else {
		tmp = (t * z) * (-4.5 / a);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+240)) {
		tmp = ((t / a) * -4.5) * z;
	} else {
		tmp = (t * z) * (-4.5 / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+240):
		tmp = ((t / a) * -4.5) * z
	else:
		tmp = (t * z) * (-4.5 / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+240))
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	else
		tmp = Float64(Float64(t * z) * Float64(-4.5 / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+240)))
		tmp = ((t / a) * -4.5) * z;
	else
		tmp = (t * z) * (-4.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+240]], $MachinePrecision]], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < -inf.0 or 2.00000000000000003e240 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 85.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 87.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 87.7

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 87.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 87.7

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 55.7

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

   9. Applied rewrites 55.7%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 2.00000000000000003e240

   1. Initial program 98.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 98.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 98.6

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 98.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 50.6

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

   9. Applied rewrites 50.6%

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

   11. Applied rewrites 57.8%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -\infty \lor \neg \left(\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 2 \cdot 10^{+240}\right):\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+27} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* x y) -5e+27) (not (<= (* x y) 5e-6)))
   (* (* (/ x a) 0.5) y)
   (* (* t z) (/ -4.5 a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -5e+27) || !((x * y) <= 5e-6)) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = (t * z) * (-4.5 / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((x * y) <= (-5d+27)) .or. (.not. ((x * y) <= 5d-6))) then
        tmp = ((x / a) * 0.5d0) * y
    else
        tmp = (t * z) * ((-4.5d0) / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -5e+27) || !((x * y) <= 5e-6)) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = (t * z) * (-4.5 / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) <= -5e+27) or not ((x * y) <= 5e-6):
		tmp = ((x / a) * 0.5) * y
	else:
		tmp = (t * z) * (-4.5 / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+27) || !(Float64(x * y) <= 5e-6))
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	else
		tmp = Float64(Float64(t * z) * Float64(-4.5 / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) <= -5e+27) || ~(((x * y) <= 5e-6)))
		tmp = ((x / a) * 0.5) * y;
	else
		tmp = (t * z) * (-4.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+27], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-6]], $MachinePrecision]], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999979e27 or 5.00000000000000041e-6 < (*.f64 x y)

   1. Initial program 90.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   if -4.99999999999999979e27 < (*.f64 x y) < 5.00000000000000041e-6

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 96.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 96.2

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 96.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 72.7

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

   9. Applied rewrites 72.7%

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

   11. Applied rewrites 78.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+27} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{a} \cdot \left(0.5 \cdot x\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t \cdot \left(-4.5 \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+24)
   (* (/ y a) (* 0.5 x))
   (if (<= (* x y) 5e-6) (/ (* t (* -4.5 z)) a) (/ (* (* x y) 0.5) a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * (-4.5 * z)) / a;
	} else {
		tmp = ((x * y) * 0.5) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+24)) then
        tmp = (y / a) * (0.5d0 * x)
    else if ((x * y) <= 5d-6) then
        tmp = (t * ((-4.5d0) * z)) / a
    else
        tmp = ((x * y) * 0.5d0) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * (-4.5 * z)) / a;
	} else {
		tmp = ((x * y) * 0.5) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+24:
		tmp = (y / a) * (0.5 * x)
	elif (x * y) <= 5e-6:
		tmp = (t * (-4.5 * z)) / a
	else:
		tmp = ((x * y) * 0.5) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+24)
		tmp = Float64(Float64(y / a) * Float64(0.5 * x));
	elseif (Float64(x * y) <= 5e-6)
		tmp = Float64(Float64(t * Float64(-4.5 * z)) / a);
	else
		tmp = Float64(Float64(Float64(x * y) * 0.5) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+24)
		tmp = (y / a) * (0.5 * x);
	elseif ((x * y) <= 5e-6)
		tmp = (t * (-4.5 * z)) / a;
	else
		tmp = ((x * y) * 0.5) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+24], N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-6], N[(N[(t * N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e24

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 84.7%

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

   if -2e24 < (*.f64 x y) < 5.00000000000000041e-6

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. 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 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 79.1%

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

   if 5.00000000000000041e-6 < (*.f64 x y)

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 81.2%

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

Alternative 5: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{a} \cdot \left(0.5 \cdot x\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t \cdot \left(-4.5 \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+24)
   (* (/ y a) (* 0.5 x))
   (if (<= (* x y) 5e-6) (/ (* t (* -4.5 z)) a) (* (* (/ x a) 0.5) y))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * (-4.5 * z)) / a;
	} else {
		tmp = ((x / a) * 0.5) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+24)) then
        tmp = (y / a) * (0.5d0 * x)
    else if ((x * y) <= 5d-6) then
        tmp = (t * ((-4.5d0) * z)) / a
    else
        tmp = ((x / a) * 0.5d0) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * (-4.5 * z)) / a;
	} else {
		tmp = ((x / a) * 0.5) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+24:
		tmp = (y / a) * (0.5 * x)
	elif (x * y) <= 5e-6:
		tmp = (t * (-4.5 * z)) / a
	else:
		tmp = ((x / a) * 0.5) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+24)
		tmp = Float64(Float64(y / a) * Float64(0.5 * x));
	elseif (Float64(x * y) <= 5e-6)
		tmp = Float64(Float64(t * Float64(-4.5 * z)) / a);
	else
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+24)
		tmp = (y / a) * (0.5 * x);
	elseif ((x * y) <= 5e-6)
		tmp = (t * (-4.5 * z)) / a;
	else
		tmp = ((x / a) * 0.5) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+24], N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-6], N[(N[(t * N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e24

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 84.7%

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

   if -2e24 < (*.f64 x y) < 5.00000000000000041e-6

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. 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 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 79.1%

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

   if 5.00000000000000041e-6 < (*.f64 x y)

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 6: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{a} \cdot \left(0.5 \cdot x\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+24)
   (* (/ y a) (* 0.5 x))
   (if (<= (* x y) 5e-6) (* (* t z) (/ -4.5 a)) (* (* (/ x a) 0.5) y))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * z) * (-4.5 / a);
	} else {
		tmp = ((x / a) * 0.5) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d+24)) then
        tmp = (y / a) * (0.5d0 * x)
    else if ((x * y) <= 5d-6) then
        tmp = (t * z) * ((-4.5d0) / a)
    else
        tmp = ((x / a) * 0.5d0) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+24) {
		tmp = (y / a) * (0.5 * x);
	} else if ((x * y) <= 5e-6) {
		tmp = (t * z) * (-4.5 / a);
	} else {
		tmp = ((x / a) * 0.5) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+24:
		tmp = (y / a) * (0.5 * x)
	elif (x * y) <= 5e-6:
		tmp = (t * z) * (-4.5 / a)
	else:
		tmp = ((x / a) * 0.5) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+24)
		tmp = Float64(Float64(y / a) * Float64(0.5 * x));
	elseif (Float64(x * y) <= 5e-6)
		tmp = Float64(Float64(t * z) * Float64(-4.5 / a));
	else
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+24)
		tmp = (y / a) * (0.5 * x);
	elseif ((x * y) <= 5e-6)
		tmp = (t * z) * (-4.5 / a);
	else
		tmp = ((x / a) * 0.5) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+24], N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-6], N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e24

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 84.7%

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

   if -2e24 < (*.f64 x y) < 5.00000000000000041e-6

   1. Initial program 96.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 96.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 96.2

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 96.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   8. 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 73.2

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

   9. Applied rewrites 73.2%

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

   11. Applied rewrites 79.0%

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

   if 5.00000000000000041e-6 < (*.f64 x y)

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 7: 91.7% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (/ (fma (* -9.0 z) t (* y x)) (* a 2.0)))

C

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

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a * 2.0))
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 94.8

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.8

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

4. Applied rewrites 94.8%

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

Alternative 8: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \mathsf{fma}\left(t, z \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (* (fma t (* z -9.0) (* x y)) (/ 0.5 a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return fma(t, (z * -9.0), (x * y)) * (0.5 / a);
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(fma(t, Float64(z * -9.0), Float64(x * y)) * Float64(0.5 / a))
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 94.8

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.8

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

4. Applied rewrites 94.8%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 94.7

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 94.7

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 94.7

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

6. Applied rewrites 94.7%

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

Alternative 9: 50.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 94.8

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.8

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

4. Applied rewrites 94.8%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 94.7

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 94.7

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 94.7

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

6. Applied rewrites 94.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
8. 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 52.4

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

9. Applied rewrites 52.4%

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

11. Applied rewrites 53.7%

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

Developer Target 1: 93.5% 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 2024307 
(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)))