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

Percentage Accurate: 91.4% → 96.1%

Time: 10.2s

Alternatives: 7

Speedup: 0.6×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \left(t \cdot -4.5\right) \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{t \cdot -4.5}{\frac{a}{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -4e+303)
     (fma (/ x 2.0) (/ y a) (* (* t -4.5) (/ z a)))
     (if (<= t_1 5e+217)
       (+ (* (* x y) (/ 0.5 a)) (* (/ 0.5 a) (* z (* t -9.0))))
       (fma (/ x 2.0) (/ y a) (/ (* t -4.5) (/ a z)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -4e+303) {
		tmp = fma((x / 2.0), (y / a), ((t * -4.5) * (z / a)));
	} else if (t_1 <= 5e+217) {
		tmp = ((x * y) * (0.5 / a)) + ((0.5 / a) * (z * (t * -9.0)));
	} else {
		tmp = fma((x / 2.0), (y / a), ((t * -4.5) / (a / z)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -4e+303)
		tmp = fma(Float64(x / 2.0), Float64(y / a), Float64(Float64(t * -4.5) * Float64(z / a)));
	elseif (t_1 <= 5e+217)
		tmp = Float64(Float64(Float64(x * y) * Float64(0.5 / a)) + Float64(Float64(0.5 / a) * Float64(z * Float64(t * -9.0))));
	else
		tmp = fma(Float64(x / 2.0), Float64(y / a), Float64(Float64(t * -4.5) / Float64(a / z)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+303], N[(N[(x / 2.0), $MachinePrecision] * N[(y / a), $MachinePrecision] + N[(N[(t * -4.5), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+217], N[(N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / a), $MachinePrecision] * N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / 2.0), $MachinePrecision] * N[(y / a), $MachinePrecision] + N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4e303

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*l* 70.7%

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

   3. Simplified 70.7%

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

      1. div-sub 67.6%

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

      2. sub-neg 67.6%

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

      3. *-commutative 67.6%

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

      4. times-frac 88.3%

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

      5. div-inv 88.3%

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

      6. associate-*r* 88.3%

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

      7. *-commutative 88.3%

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

      8. associate-*l* 88.3%

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

      9. *-commutative 88.3%

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

      10. associate-/r* 88.3%

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

      11. metadata-eval 88.3%

          \[\leadsto \frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{\color{blue}{0.5}}{a}\right) \]

   5. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]
   6. Step-by-step derivation

      1. fma-def 88.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]

      2. associate-*r/ 88.3%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\color{blue}{\frac{\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      3. distribute-neg-frac 88.3%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \color{blue}{\frac{-\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      4. *-commutative 88.3%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(\left(z \cdot t\right) \cdot 9\right)} \cdot 0.5}{a}\right) \]

      5. associate-*l* 88.3%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(z \cdot t\right) \cdot \left(9 \cdot 0.5\right)}}{a}\right) \]

      6. distribute-rgt-neg-in 88.3%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{\color{blue}{\left(z \cdot t\right) \cdot \left(-9 \cdot 0.5\right)}}{a}\right) \]

      7. metadata-eval 88.3%

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

      8. metadata-eval 88.3%

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

      9. associate-*l/ 88.3%

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

      10. *-commutative 88.3%

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

      11. associate-/l* 96.7%

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

      12. associate-*l/ 96.7%

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

   7. Simplified 96.7%

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

      1. div-inv 96.6%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \color{blue}{\left(t \cdot -4.5\right) \cdot \frac{1}{\frac{a}{z}}}\right) \]

      2. clear-num 96.7%

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

   9. Applied egg-rr 96.7%

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

   if -4e303 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 5.00000000000000041e217

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in a around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. metadata-eval 99.1%

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

      4. +-commutative 99.1%

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

      5. associate-/l* 98.4%

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

      6. +-commutative 98.4%

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

      7. metadata-eval 98.4%

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

      8. cancel-sign-sub-inv 98.4%

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

      9. fma-neg 98.4%

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

      10. *-commutative 98.4%

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

      11. distribute-lft-neg-in 98.4%

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

      12. metadata-eval 98.4%

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

      13. *-commutative 98.4%

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

      14. associate-*l* 98.4%

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

   6. Simplified 98.4%

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

      1. associate-/r/ 99.1%

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

      2. associate-*r* 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. *-commutative 99.1%

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

      6. fma-def 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. *-commutative 99.1%

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

      9. distribute-rgt-neg-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-*r* 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 5.00000000000000041e217 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*l* 78.3%

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

   3. Simplified 78.3%

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

      1. div-sub 68.8%

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

      2. sub-neg 68.8%

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

      3. *-commutative 68.8%

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

      4. times-frac 81.3%

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

      5. div-inv 81.2%

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

      6. associate-*r* 81.2%

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

      7. *-commutative 81.2%

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

      8. associate-*l* 81.3%

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

      9. *-commutative 81.3%

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

      10. associate-/r* 81.3%

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

      11. metadata-eval 81.3%

          \[\leadsto \frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{\color{blue}{0.5}}{a}\right) \]

   5. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]
   6. Step-by-step derivation

      1. fma-def 83.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]

      2. associate-*r/ 83.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\color{blue}{\frac{\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      3. distribute-neg-frac 83.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \color{blue}{\frac{-\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      4. *-commutative 83.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(\left(z \cdot t\right) \cdot 9\right)} \cdot 0.5}{a}\right) \]

      5. associate-*l* 83.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(z \cdot t\right) \cdot \left(9 \cdot 0.5\right)}}{a}\right) \]

      6. distribute-rgt-neg-in 83.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{\color{blue}{\left(z \cdot t\right) \cdot \left(-9 \cdot 0.5\right)}}{a}\right) \]

      7. metadata-eval 83.2%

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

      8. metadata-eval 83.2%

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

      9. associate-*l/ 83.2%

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

      10. *-commutative 83.2%

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

      11. associate-/l* 88.5%

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

      12. associate-*l/ 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \left(t \cdot -4.5\right) \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{t \cdot -4.5}{\frac{a}{z}}\right)\\ \end{array} \]

Alternative 2: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+303} \lor \neg \left(t_1 \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \left(t \cdot -4.5\right) \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -4e+303) (not (<= t_1 5e+262)))
     (fma (/ x 2.0) (/ y a) (* (* t -4.5) (/ z a)))
     (+ (* (* x y) (/ 0.5 a)) (* (/ 0.5 a) (* z (* t -9.0)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -4e+303) || !(t_1 <= 5e+262)) {
		tmp = fma((x / 2.0), (y / a), ((t * -4.5) * (z / a)));
	} else {
		tmp = ((x * y) * (0.5 / a)) + ((0.5 / a) * (z * (t * -9.0)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -4e+303) || !(t_1 <= 5e+262))
		tmp = fma(Float64(x / 2.0), Float64(y / a), Float64(Float64(t * -4.5) * Float64(z / a)));
	else
		tmp = Float64(Float64(Float64(x * y) * Float64(0.5 / a)) + Float64(Float64(0.5 / a) * Float64(z * Float64(t * -9.0))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+303], N[Not[LessEqual[t$95$1, 5e+262]], $MachinePrecision]], N[(N[(x / 2.0), $MachinePrecision] * N[(y / a), $MachinePrecision] + N[(N[(t * -4.5), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / a), $MachinePrecision] * N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4e303 or 5.00000000000000008e262 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

      1. div-sub 65.6%

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

      2. sub-neg 65.6%

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

      3. *-commutative 65.6%

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

      4. times-frac 82.5%

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

      5. div-inv 82.5%

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

      6. associate-*r* 82.5%

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

      7. *-commutative 82.5%

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

      8. associate-*l* 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-/r* 82.5%

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

      11. metadata-eval 82.5%

          \[\leadsto \frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{\color{blue}{0.5}}{a}\right) \]

   5. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot \frac{y}{a} + \left(-\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]
   6. Step-by-step derivation

      1. fma-def 83.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]

      2. associate-*r/ 83.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, -\color{blue}{\frac{\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      3. distribute-neg-frac 83.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \color{blue}{\frac{-\left(9 \cdot \left(z \cdot t\right)\right) \cdot 0.5}{a}}\right) \]

      4. *-commutative 83.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(\left(z \cdot t\right) \cdot 9\right)} \cdot 0.5}{a}\right) \]

      5. associate-*l* 83.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{-\color{blue}{\left(z \cdot t\right) \cdot \left(9 \cdot 0.5\right)}}{a}\right) \]

      6. distribute-rgt-neg-in 83.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \frac{\color{blue}{\left(z \cdot t\right) \cdot \left(-9 \cdot 0.5\right)}}{a}\right) \]

      7. metadata-eval 83.8%

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

      8. metadata-eval 83.8%

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

      9. associate-*l/ 83.8%

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

      10. *-commutative 83.8%

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

      11. associate-/l* 90.9%

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

      12. associate-*l/ 90.8%

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

   7. Simplified 90.8%

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

      1. div-inv 90.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \color{blue}{\left(t \cdot -4.5\right) \cdot \frac{1}{\frac{a}{z}}}\right) \]

      2. clear-num 90.8%

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

   9. Applied egg-rr 90.8%

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

   if -4e303 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 5.00000000000000008e262

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in a around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. metadata-eval 99.1%

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

      4. +-commutative 99.1%

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

      5. associate-/l* 98.4%

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

      6. +-commutative 98.4%

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

      7. metadata-eval 98.4%

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

      8. cancel-sign-sub-inv 98.4%

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

      9. fma-neg 98.5%

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

      10. *-commutative 98.5%

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

      11. distribute-lft-neg-in 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-*l* 98.4%

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

   6. Simplified 98.4%

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

      1. associate-/r/ 99.1%

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

      2. associate-*r* 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. *-commutative 99.1%

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

      6. fma-def 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. *-commutative 99.1%

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

      9. distribute-rgt-neg-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. associate-*r* 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+303} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{2}, \frac{y}{a}, \left(t \cdot -4.5\right) \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{0.5}{a} \cdot \left(z \cdot \left(t \cdot -9\right)\right)\\ \end{array} \]

Alternative 3: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+266} \lor \neg \left(t_1 \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;z \cdot \left(\frac{0.5}{a} \cdot \left(t \cdot -9\right)\right) + x \cdot \frac{y}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -2e+266) (not (<= t_1 2e+307)))
     (+ (* z (* (/ 0.5 a) (* t -9.0))) (* x (/ y (* 2.0 a))))
     (/ t_1 (* 2.0 a)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -2e+266) || !(t_1 <= 2e+307)) {
		tmp = (z * ((0.5 / a) * (t * -9.0))) + (x * (y / (2.0 * a)));
	} else {
		tmp = t_1 / (2.0 * a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - ((z * 9.0d0) * t)
    if ((t_1 <= (-2d+266)) .or. (.not. (t_1 <= 2d+307))) then
        tmp = (z * ((0.5d0 / a) * (t * (-9.0d0)))) + (x * (y / (2.0d0 * a)))
    else
        tmp = t_1 / (2.0d0 * a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -2e+266) || !(t_1 <= 2e+307)) {
		tmp = (z * ((0.5 / a) * (t * -9.0))) + (x * (y / (2.0 * a)));
	} else {
		tmp = t_1 / (2.0 * a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) - ((z * 9.0) * t)
	tmp = 0
	if (t_1 <= -2e+266) or not (t_1 <= 2e+307):
		tmp = (z * ((0.5 / a) * (t * -9.0))) + (x * (y / (2.0 * a)))
	else:
		tmp = t_1 / (2.0 * a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= -2e+266) || !(t_1 <= 2e+307))
		tmp = Float64(Float64(z * Float64(Float64(0.5 / a) * Float64(t * -9.0))) + Float64(x * Float64(y / Float64(2.0 * a))));
	else
		tmp = Float64(t_1 / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - ((z * 9.0) * t);
	tmp = 0.0;
	if ((t_1 <= -2e+266) || ~((t_1 <= 2e+307)))
		tmp = (z * ((0.5 / a) * (t * -9.0))) + (x * (y / (2.0 * a)));
	else
		tmp = t_1 / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+266], N[Not[LessEqual[t$95$1, 2e+307]], $MachinePrecision]], N[(N[(z * N[(N[(0.5 / a), $MachinePrecision] * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -2.0000000000000001e266 or 1.99999999999999997e307 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*l* 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in a around 0 70.7%

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

      1. associate-*r/ 70.7%

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

      2. cancel-sign-sub-inv 70.7%

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

      3. metadata-eval 70.7%

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

      4. +-commutative 70.7%

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

      5. associate-/l* 70.7%

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

      6. +-commutative 70.7%

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

      7. metadata-eval 70.7%

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

      8. cancel-sign-sub-inv 70.7%

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

      9. fma-neg 72.1%

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

      10. *-commutative 72.1%

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

      11. distribute-lft-neg-in 72.1%

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

      12. metadata-eval 72.1%

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

      13. *-commutative 72.1%

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

      14. associate-*l* 72.1%

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

   6. Simplified 72.1%

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

      1. add-cube-cbrt 72.1%

         \[\leadsto \frac{0.5}{\color{blue}{\left(\sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}} \cdot \sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}\right) \cdot \sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}}} \]

      2. pow3 72.1%

         \[\leadsto \frac{0.5}{\color{blue}{{\left(\sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}\right)}^{3}}} \]

   8. Applied egg-rr 72.1%

      \[\leadsto \frac{0.5}{\color{blue}{{\left(\sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}\right)}^{3}}} \]
   9. Step-by-step derivation

      1. unpow3 72.1%

         \[\leadsto \frac{0.5}{\color{blue}{\left(\sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}} \cdot \sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}\right) \cdot \sqrt[3]{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}}} \]

      2. add-cube-cbrt 72.1%

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

      3. associate-/r/ 72.1%

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

      4. fma-udef 70.7%

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

      5. +-commutative 70.7%

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

      6. distribute-rgt-out 66.5%

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

      7. associate-*l* 75.6%

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

      8. associate-*r* 94.1%

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

      9. clear-num 94.1%

         \[\leadsto z \cdot \left(\left(t \cdot -9\right) \cdot \frac{0.5}{a}\right) + x \cdot \left(y \cdot \color{blue}{\frac{1}{\frac{a}{0.5}}}\right) \]

      10. un-div-inv 94.1%

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

      11. div-inv 94.1%

          \[\leadsto z \cdot \left(\left(t \cdot -9\right) \cdot \frac{0.5}{a}\right) + x \cdot \frac{y}{\color{blue}{a \cdot \frac{1}{0.5}}} \]

      12. metadata-eval 94.1%

          \[\leadsto z \cdot \left(\left(t \cdot -9\right) \cdot \frac{0.5}{a}\right) + x \cdot \frac{y}{a \cdot \color{blue}{2}} \]

   10. Applied egg-rr 94.1%

       \[\leadsto \color{blue}{z \cdot \left(\left(t \cdot -9\right) \cdot \frac{0.5}{a}\right) + x \cdot \frac{y}{a \cdot 2}} \]

   if -2.0000000000000001e266 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.99999999999999997e307

   1. Initial program 98.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+266} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+307}\right):\\ \;\;\;\;z \cdot \left(\frac{0.5}{a} \cdot \left(t \cdot -9\right)\right) + x \cdot \frac{y}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2 \cdot a}\\ \end{array} \]

Alternative 4: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(z \cdot t\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* 0.5 (/ x (/ a y)))
   (if (<= (* x y) 5e+217)
     (/ (- (* x y) (* 9.0 (* z t))) (* 2.0 a))
     (* 0.5 (* x (/ y a))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = 0.5 * (x / (a / y));
	} else if ((x * y) <= 5e+217) {
		tmp = ((x * y) - (9.0 * (z * t))) / (2.0 * a);
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (x / (a / y));
	} else if ((x * y) <= 5e+217) {
		tmp = ((x * y) - (9.0 * (z * t))) / (2.0 * a);
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = 0.5 * (x / (a / y))
	elif (x * y) <= 5e+217:
		tmp = ((x * y) - (9.0 * (z * t))) / (2.0 * a)
	else:
		tmp = 0.5 * (x * (y / a))
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(x / Float64(a / y)));
	elseif (Float64(x * y) <= 5e+217)
		tmp = Float64(Float64(Float64(x * y) - Float64(9.0 * Float64(z * t))) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = 0.5 * (x / (a / y));
	elseif ((x * y) <= 5e+217)
		tmp = ((x * y) - (9.0 * (z * t))) / (2.0 * a);
	else
		tmp = 0.5 * (x * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(0.5 * N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+217], N[(N[(N[(x * y), $MachinePrecision] - N[(9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.5%

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

      1. *-commutative 58.5%

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

      2. *-commutative 58.5%

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

      3. associate-*l* 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in x around inf 58.5%

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

      1. associate-/l* 94.1%

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

   6. Simplified 94.1%

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

   if -inf.0 < (*.f64 x y) < 5.00000000000000041e217

   1. Initial program 96.1%

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

      1. fma-neg 96.1%

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

      2. associate-*l* 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. *-commutative 96.1%

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

      5. distribute-rgt-neg-in 96.1%

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

      6. metadata-eval 96.1%

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

   3. Simplified 96.1%

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

      1. *-commutative 96.1%

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

      2. metadata-eval 96.1%

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

      3. distribute-lft-neg-in 96.1%

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

      4. distribute-rgt-neg-in 96.1%

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

      5. fma-neg 96.1%

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

      6. associate-*r* 96.1%

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

      7. *-commutative 96.1%

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

      8. associate-*l* 96.1%

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

   5. Applied egg-rr 96.1%

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

   if 5.00000000000000041e217 < (*.f64 x y)

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. *-commutative 73.1%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in x around inf 73.2%

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

      1. associate-*r/ 93.7%

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

   6. Simplified 93.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(z \cdot t\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]

Alternative 5: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -500000000 \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-87}\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* x y) -500000000.0) (not (<= (* x y) 2e-87)))
   (* y (* 0.5 (/ x a)))
   (* -4.5 (/ (* z t) a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -500000000.0) || !((x * y) <= 2e-87)) {
		tmp = y * (0.5 * (x / a));
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t 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) <= (-500000000.0d0)) .or. (.not. ((x * y) <= 2d-87))) then
        tmp = y * (0.5d0 * (x / a))
    else
        tmp = (-4.5d0) * ((z * t) / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) <= -500000000.0) || !((x * y) <= 2e-87)) {
		tmp = y * (0.5 * (x / a));
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) <= -500000000.0) or not ((x * y) <= 2e-87):
		tmp = y * (0.5 * (x / a))
	else:
		tmp = -4.5 * ((z * t) / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(x * y) <= -500000000.0) || !(Float64(x * y) <= 2e-87))
		tmp = Float64(y * Float64(0.5 * Float64(x / a)));
	else
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) <= -500000000.0) || ~(((x * y) <= 2e-87)))
		tmp = y * (0.5 * (x / a));
	else
		tmp = -4.5 * ((z * t) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -500000000.0], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-87]], $MachinePrecision]], N[(y * N[(0.5 * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e8 or 2.00000000000000004e-87 < (*.f64 x y)

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*l* 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in x around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*r* 68.3%

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

   6. Simplified 68.3%

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

      1. associate-/l* 72.0%

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

      2. associate-/r/ 72.7%

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

      3. *-commutative 72.7%

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

      4. associate-*l/ 72.7%

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

      5. metadata-eval 72.7%

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

      6. div-inv 72.7%

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

      7. *-commutative 72.7%

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

      8. associate-/r* 72.7%

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

      9. associate-*r/ 67.7%

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

      10. associate-*l/ 71.4%

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

      11. *-un-lft-identity 71.4%

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

      12. *-commutative 71.4%

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

      13. times-frac 72.0%

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

      14. metadata-eval 72.0%

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

   8. Applied egg-rr 72.0%

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

   if -5e8 < (*.f64 x y) < 2.00000000000000004e-87

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*l* 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in x around 0 81.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -500000000 \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-87}\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]

Alternative 6: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.56e-133)
   (* -4.5 (/ (* z t) a))
   (if (<= t 4e+49) (* 0.5 (* x (/ y a))) (* -4.5 (* z (/ t a))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.56e-133) {
		tmp = -4.5 * ((z * t) / a);
	} else if (t <= 4e+49) {
		tmp = 0.5 * (x * (y / a));
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.56d-133)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (t <= 4d+49) then
        tmp = 0.5d0 * (x * (y / a))
    else
        tmp = (-4.5d0) * (z * (t / a))
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.56e-133) {
		tmp = -4.5 * ((z * t) / a);
	} else if (t <= 4e+49) {
		tmp = 0.5 * (x * (y / a));
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.56e-133:
		tmp = -4.5 * ((z * t) / a)
	elif t <= 4e+49:
		tmp = 0.5 * (x * (y / a))
	else:
		tmp = -4.5 * (z * (t / a))
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.56e-133)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif (t <= 4e+49)
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.56e-133)
		tmp = -4.5 * ((z * t) / a);
	elseif (t <= 4e+49)
		tmp = 0.5 * (x * (y / a));
	else
		tmp = -4.5 * (z * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.56e-133], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+49], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.56e-133

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. *-commutative 88.6%

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

      3. associate-*l* 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 62.2%

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

   if -1.56e-133 < t < 3.99999999999999979e49

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-*l* 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in x around inf 74.5%

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

      1. associate-*r/ 76.0%

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

   6. Simplified 76.0%

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

   if 3.99999999999999979e49 < t

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*l* 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 73.9%

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

      1. associate-/l* 73.6%

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

   6. Simplified 73.6%

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

      1. associate-/r/ 81.0%

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

   8. Applied egg-rr 81.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.56 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 7: 50.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. *-commutative 90.9%

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

   2. *-commutative 90.9%

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

   3. associate-*l* 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in x around 0 50.7%

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

   1. associate-/l* 48.9%

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

6. Simplified 48.9%

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

   1. associate-/r/ 50.6%

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

8. Applied egg-rr 50.6%

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

9. Final simplification 50.6%

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

Developer target: 93.7% accurate, 0.7× 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 2023336 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

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

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