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

Percentage Accurate: 90.9% → 94.8%

Time: 13.0s

Alternatives: 11

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% 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: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t}{a} \cdot \left(z \cdot -4.5\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{x \cdot y - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

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 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (/ t a) (* z -4.5))
     (if (<= t_1 5e+280)
       (/ (- (* x y) t_1) (* a 2.0))
       (* -4.5 (/ t (/ a z)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t / a) * (z * -4.5);
	} else if (t_1 <= 5e+280) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = -4.5 * (t / (a / z));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (t / a) * (z * -4.5);
	} else if (t_1 <= 5e+280) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = -4.5 * (t / (a / z));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (t / a) * (z * -4.5)
	elif t_1 <= 5e+280:
		tmp = ((x * y) - t_1) / (a * 2.0)
	else:
		tmp = -4.5 * (t / (a / z))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t / a) * Float64(z * -4.5));
	elseif (t_1 <= 5e+280)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a * 2.0));
	else
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (t / a) * (z * -4.5);
	elseif (t_1 <= 5e+280)
		tmp = ((x * y) - t_1) / (a * 2.0);
	else
		tmp = -4.5 * (t / (a / z));
	end
	tmp_2 = tmp;
end

Wolfram

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[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t / a), $MachinePrecision] * N[(z * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+280], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z 9) t) < -inf.0

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. +-commutative 63.3%

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

      3. associate-*l* 63.3%

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

      4. distribute-rgt-neg-in 63.3%

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

      5. fma-def 63.3%

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

      6. *-commutative 63.3%

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

      7. distribute-rgt-neg-in 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

      1. add-cube-cbrt 63.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}} \]

      2. pow3 63.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right)}^{3}} \]

      3. div-inv 63.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot 2}}}\right)}^{3} \]

      4. metadata-eval 63.3%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}}}\right)}^{3} \]

      5. div-inv 63.3%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}}}\right)}^{3} \]

      6. clear-num 63.3%

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

   5. Applied egg-rr 63.3%

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

   6. Taylor expanded in z around inf 68.9%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-4.5 \cdot \frac{t \cdot z}{a}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. *-commutative 68.9%

         \[\leadsto {\left(\sqrt[3]{-4.5 \cdot \frac{\color{blue}{z \cdot t}}{a}}\right)}^{3} \]

      2. associate-*r/ 94.1%

         \[\leadsto {\left(\sqrt[3]{-4.5 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)}}\right)}^{3} \]

   8. Simplified 94.1%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-4.5 \cdot \left(z \cdot \frac{t}{a}\right)}}\right)}^{3} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 94.6%

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

      2. associate-*r* 94.8%

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

   10. Applied egg-rr 94.8%

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

   if -inf.0 < (*.f64 (*.f64 z 9) t) < 5.0000000000000002e280

   1. Initial program 96.2%

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

   if 5.0000000000000002e280 < (*.f64 (*.f64 z 9) t)

   1. Initial program 52.1%

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

      1. div-sub 52.1%

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

      2. +-rgt-identity 52.1%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 52.1%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 52.1%

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

      5. fma-neg 59.2%

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

      6. associate-*l* 59.2%

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

      7. distribute-rgt-neg-in 59.2%

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

      8. *-commutative 59.2%

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

      9. distribute-rgt-neg-in 59.2%

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

      10. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 59.3%

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

      1. associate-/l* 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\frac{t}{a} \cdot \left(z \cdot -4.5\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 2: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{0}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

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) (* (* z 9.0) t)) -4e+277)
   (fma -4.5 (* z (/ t a)) (fma 0.5 (/ y (/ a x)) (/ 0.0 t)))
   (/ (fma x y (* z (* t -9.0))) (* a 2.0))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= -4e+277) {
		tmp = fma(-4.5, (z * (t / a)), fma(0.5, (y / (a / x)), (0.0 / t)));
	} else {
		tmp = fma(x, y, (z * (t * -9.0))) / (a * 2.0);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= -4e+277)
		tmp = fma(-4.5, Float64(z * Float64(t / a)), fma(0.5, Float64(y / Float64(a / x)), Float64(0.0 / t)));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: z and t 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+277], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision] + N[(0.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4.00000000000000001e277

   1. Initial program 63.7%

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

      1. sub-neg 63.7%

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

      2. +-commutative 63.7%

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

      3. neg-sub0 63.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 63.7%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 63.7%

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

      6. neg-mul-1 63.7%

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

      7. associate-/l* 63.7%

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

      8. associate-/r/ 63.7%

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

      9. *-commutative 63.7%

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

      10. sub-neg 63.7%

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

      11. +-commutative 63.7%

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

      12. neg-sub0 63.7%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 63.7%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 63.7%

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

      15. distribute-lft-neg-out 63.7%

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

      16. distribute-rgt-neg-in 63.7%

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

   3. Simplified 63.8%

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

   5. Applied egg-rr 4.0%

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

   6. Taylor expanded in t around inf 19.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + \left(0.5 \cdot \frac{y \cdot x}{a} + -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right)} \]
   7. Step-by-step derivation

      1. fma-def 19.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5, \frac{t \cdot z}{a}, 0.5 \cdot \frac{y \cdot x}{a} + -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right)} \]

      2. *-commutative 19.8%

         \[\leadsto \mathsf{fma}\left(-4.5, \frac{\color{blue}{z \cdot t}}{a}, 0.5 \cdot \frac{y \cdot x}{a} + -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right) \]

      3. associate-*r/ 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, \color{blue}{z \cdot \frac{t}{a}}, 0.5 \cdot \frac{y \cdot x}{a} + -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right) \]

      4. fma-def 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \color{blue}{\mathsf{fma}\left(0.5, \frac{y \cdot x}{a}, -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right)}\right) \]

      5. associate-/l* 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \color{blue}{\frac{y}{\frac{a}{x}}}, -1 \cdot \frac{0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}}{t}\right)\right) \]

      6. associate-*r/ 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \color{blue}{\frac{-1 \cdot \left(0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} + -0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot z}\right)}{t}}\right)\right) \]

      7. distribute-rgt-out 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{-1 \cdot \color{blue}{\left(\frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} \cdot \left(0.05555555555555555 + -0.05555555555555555\right)\right)}}{t}\right)\right) \]

      8. metadata-eval 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{-1 \cdot \left(\frac{{y}^{2} \cdot {x}^{2}}{a \cdot z} \cdot \color{blue}{0}\right)}{t}\right)\right) \]

      9. mul0-rgt 92.0%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{-1 \cdot \color{blue}{0}}{t}\right)\right) \]

      10. metadata-eval 92.0%

          \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{\color{blue}{0}}{t}\right)\right) \]

   8. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{0}{t}\right)\right)} \]

   if -4.00000000000000001e277 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 94.7%

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

      1. div-sub 93.8%

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

      2. +-rgt-identity 93.8%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 94.7%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 94.7%

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

      5. fma-neg 95.5%

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

      6. associate-*l* 95.5%

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

      7. distribute-rgt-neg-in 95.5%

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

      8. *-commutative 95.5%

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

      9. distribute-rgt-neg-in 95.5%

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

      10. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \mathsf{fma}\left(0.5, \frac{y}{\frac{a}{x}}, \frac{0}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternative 3: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, y \cdot \left(x \cdot \frac{0.5}{a}\right)\right) + \frac{0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

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) (* (* z 9.0) t)) -4e+277)
   (+ (fma -4.5 (* z (/ t a)) (* y (* x (/ 0.5 a)))) (/ 0.0 z))
   (/ (fma x y (* z (* t -9.0))) (* a 2.0))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= -4e+277) {
		tmp = fma(-4.5, (z * (t / a)), (y * (x * (0.5 / a)))) + (0.0 / z);
	} else {
		tmp = fma(x, y, (z * (t * -9.0))) / (a * 2.0);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= -4e+277)
		tmp = Float64(fma(-4.5, Float64(z * Float64(t / a)), Float64(y * Float64(x * Float64(0.5 / a)))) + Float64(0.0 / z));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

NOTE: z and t 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+277], N[(N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 / z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4.00000000000000001e277

   1. Initial program 63.7%

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

      1. sub-neg 63.7%

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

      2. +-commutative 63.7%

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

      3. neg-sub0 63.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 63.7%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 63.7%

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

      6. neg-mul-1 63.7%

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

      7. associate-/l* 63.7%

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

      8. associate-/r/ 63.7%

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

      9. *-commutative 63.7%

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

      10. sub-neg 63.7%

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

      11. +-commutative 63.7%

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

      12. neg-sub0 63.7%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 63.7%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 63.7%

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

      15. distribute-lft-neg-out 63.7%

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

      16. distribute-rgt-neg-in 63.7%

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

   3. Simplified 63.8%

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

   5. Applied egg-rr 4.0%

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

   6. Taylor expanded in z around inf 19.8%

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + \left(-1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} + 0.5 \cdot \frac{y \cdot x}{a}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 19.8%

         \[\leadsto -4.5 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(0.5 \cdot \frac{y \cdot x}{a} + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z}\right)} \]

      2. associate-+r+ 19.8%

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z}} \]

      3. fma-def 19.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5, \frac{t \cdot z}{a}, 0.5 \cdot \frac{y \cdot x}{a}\right)} + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      4. *-commutative 19.8%

         \[\leadsto \mathsf{fma}\left(-4.5, \frac{\color{blue}{z \cdot t}}{a}, 0.5 \cdot \frac{y \cdot x}{a}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      5. associate-*r/ 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, \color{blue}{z \cdot \frac{t}{a}}, 0.5 \cdot \frac{y \cdot x}{a}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      6. associate-*r/ 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \color{blue}{\frac{0.5 \cdot \left(y \cdot x\right)}{a}}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      7. *-commutative 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \frac{0.5 \cdot \color{blue}{\left(x \cdot y\right)}}{a}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      8. associate-*l/ 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \color{blue}{\frac{0.5}{a} \cdot \left(x \cdot y\right)}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      9. associate-*r* 30.6%

         \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \color{blue}{\left(\frac{0.5}{a} \cdot x\right) \cdot y}\right) + -1 \cdot \frac{-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}}{z} \]

      10. associate-*r/ 30.6%

          \[\leadsto \mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right) + \color{blue}{\frac{-1 \cdot \left(-0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t} + 0.05555555555555555 \cdot \frac{{y}^{2} \cdot {x}^{2}}{a \cdot t}\right)}{z}} \]

   8. Simplified 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right) + \frac{0}{z}} \]

   if -4.00000000000000001e277 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 94.7%

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

      1. div-sub 93.8%

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

      2. +-rgt-identity 93.8%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 94.7%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 94.7%

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

      5. fma-neg 95.5%

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

      6. associate-*l* 95.5%

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

      7. distribute-rgt-neg-in 95.5%

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

      8. *-commutative 95.5%

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

      9. distribute-rgt-neg-in 95.5%

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

      10. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, z \cdot \frac{t}{a}, y \cdot \left(x \cdot \frac{0.5}{a}\right)\right) + \frac{0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternative 4: 69.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59} \lor \neg \left(x \cdot y \leq 10^{+130}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \end{array} \]

FPCore

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) -2e-44)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 2e-133)
     (* -4.5 (/ t (/ a z)))
     (if (or (<= (* x y) 5e+59) (not (<= (* x y) 1e+130)))
       (* (* x y) (/ 0.5 a))
       (* -4.5 (* z (/ t a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = -4.5 * (t / (a / z));
	} else if (((x * y) <= 5e+59) || !((x * y) <= 1e+130)) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Fortran

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) <= (-2d-44)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 2d-133) then
        tmp = (-4.5d0) * (t / (a / z))
    else if (((x * y) <= 5d+59) .or. (.not. ((x * y) <= 1d+130))) then
        tmp = (x * y) * (0.5d0 / a)
    else
        tmp = (-4.5d0) * (z * (t / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = -4.5 * (t / (a / z));
	} else if (((x * y) <= 5e+59) || !((x * y) <= 1e+130)) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = -4.5 * (z * (t / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-44:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 2e-133:
		tmp = -4.5 * (t / (a / z))
	elif ((x * y) <= 5e+59) or not ((x * y) <= 1e+130):
		tmp = (x * y) * (0.5 / a)
	else:
		tmp = -4.5 * (z * (t / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-44)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 2e-133)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif ((Float64(x * y) <= 5e+59) || !(Float64(x * y) <= 1e+130))
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	else
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-44)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 2e-133)
		tmp = -4.5 * (t / (a / z));
	elseif (((x * y) <= 5e+59) || ~(((x * y) <= 1e+130)))
		tmp = (x * y) * (0.5 / a);
	else
		tmp = -4.5 * (z * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

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], -2e-44], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-133], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], 5e+59], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+130]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999991e-44

   1. Initial program 90.4%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.4%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.4%

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

      5. fma-neg 91.8%

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

      6. associate-*l* 91.8%

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

      7. distribute-rgt-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. distribute-rgt-neg-in 91.8%

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

      10. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

   6. Simplified 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r/ 69.9%

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

      3. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -1.99999999999999991e-44 < (*.f64 x y) < 2.0000000000000001e-133

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. +-rgt-identity 90.3%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.3%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.3%

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

      5. fma-neg 90.3%

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

      6. associate-*l* 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

   if 2.0000000000000001e-133 < (*.f64 x y) < 4.9999999999999997e59 or 1.0000000000000001e130 < (*.f64 x y)

   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. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. neg-sub0 94.1%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 94.1%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 94.1%

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

      6. neg-mul-1 94.1%

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

      7. associate-/l* 93.7%

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

      8. associate-/r/ 93.9%

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

      9. *-commutative 93.9%

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

      10. sub-neg 93.9%

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

      11. +-commutative 93.9%

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

      12. neg-sub0 93.9%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 93.9%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 93.9%

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

      15. distribute-lft-neg-out 93.9%

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

      16. distribute-rgt-neg-in 93.9%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around inf 75.4%

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

   if 4.9999999999999997e59 < (*.f64 x y) < 1.0000000000000001e130

   1. Initial program 88.8%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 88.8%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 88.8%

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

      5. fma-neg 88.8%

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

      6. associate-*l* 89.2%

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

      7. distribute-rgt-neg-in 89.2%

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

      8. *-commutative 89.2%

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

      9. distribute-rgt-neg-in 89.2%

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

      10. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 64.3%

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

      1. associate-/l* 63.9%

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

      2. associate-/r/ 74.5%

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

   6. Simplified 74.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59} \lor \neg \left(x \cdot y \leq 10^{+130}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 5: 70.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

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) -2e-44)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 2e-133)
     (* -4.5 (/ t (/ a z)))
     (if (<= (* x y) 5e+59)
       (* (* x y) (/ 0.5 a))
       (if (<= (* x y) 1e+130)
         (* -4.5 (* z (/ t a)))
         (* (* y 0.5) (/ x a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = -4.5 * (t / (a / z));
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Fortran

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) <= (-2d-44)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 2d-133) then
        tmp = (-4.5d0) * (t / (a / z))
    else if ((x * y) <= 5d+59) then
        tmp = (x * y) * (0.5d0 / a)
    else if ((x * y) <= 1d+130) then
        tmp = (-4.5d0) * (z * (t / a))
    else
        tmp = (y * 0.5d0) * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = -4.5 * (t / (a / z));
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-44:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 2e-133:
		tmp = -4.5 * (t / (a / z))
	elif (x * y) <= 5e+59:
		tmp = (x * y) * (0.5 / a)
	elif (x * y) <= 1e+130:
		tmp = -4.5 * (z * (t / a))
	else:
		tmp = (y * 0.5) * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-44)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 2e-133)
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	elseif (Float64(x * y) <= 5e+59)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	elseif (Float64(x * y) <= 1e+130)
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	else
		tmp = Float64(Float64(y * 0.5) * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-44)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 2e-133)
		tmp = -4.5 * (t / (a / z));
	elseif ((x * y) <= 5e+59)
		tmp = (x * y) * (0.5 / a);
	elseif ((x * y) <= 1e+130)
		tmp = -4.5 * (z * (t / a));
	else
		tmp = (y * 0.5) * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

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], -2e-44], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-133], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+59], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999991e-44

   1. Initial program 90.4%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.4%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.4%

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

      5. fma-neg 91.8%

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

      6. associate-*l* 91.8%

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

      7. distribute-rgt-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. distribute-rgt-neg-in 91.8%

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

      10. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

   6. Simplified 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r/ 69.9%

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

      3. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -1.99999999999999991e-44 < (*.f64 x y) < 2.0000000000000001e-133

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. +-rgt-identity 90.3%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.3%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.3%

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

      5. fma-neg 90.3%

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

      6. associate-*l* 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

   if 2.0000000000000001e-133 < (*.f64 x y) < 4.9999999999999997e59

   1. Initial program 96.9%

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

      1. sub-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. neg-sub0 96.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 96.9%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 96.9%

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

      6. neg-mul-1 96.9%

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

      7. associate-/l* 96.1%

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

      8. associate-/r/ 96.8%

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

      9. *-commutative 96.8%

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

      10. sub-neg 96.8%

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

      11. +-commutative 96.8%

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

      12. neg-sub0 96.8%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 96.8%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 96.8%

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

      15. distribute-lft-neg-out 96.8%

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

      16. distribute-rgt-neg-in 96.8%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around inf 65.3%

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

   if 4.9999999999999997e59 < (*.f64 x y) < 1.0000000000000001e130

   1. Initial program 88.8%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 88.8%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 88.8%

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

      5. fma-neg 88.8%

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

      6. associate-*l* 89.2%

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

      7. distribute-rgt-neg-in 89.2%

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

      8. *-commutative 89.2%

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

      9. distribute-rgt-neg-in 89.2%

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

      10. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 64.3%

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

      1. associate-/l* 63.9%

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

      2. associate-/r/ 74.5%

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

   6. Simplified 74.5%

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

   if 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 92.0%

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

      1. sub-neg 92.0%

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

      2. +-commutative 92.0%

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

      3. associate-*l* 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. fma-def 94.0%

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

      6. *-commutative 94.0%

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

      7. distribute-rgt-neg-in 94.0%

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

      8. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

      1. add-cube-cbrt 93.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}} \]

      2. pow3 93.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right)}^{3}} \]

      3. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot 2}}}\right)}^{3} \]

      4. metadata-eval 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}}}\right)}^{3} \]

      5. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}}}\right)}^{3} \]

      6. clear-num 93.5%

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

   5. Applied egg-rr 93.5%

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

      1. cbrt-prod 93.3%

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

   7. Applied egg-rr 93.3%

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

      1. *-commutative 93.3%

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

   9. Simplified 93.3%

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

   10. Taylor expanded in z around 0 82.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{y \cdot x}{a}\right)} \]
   11. Step-by-step derivation

       1. pow-base-1 82.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{1} \cdot \frac{y \cdot x}{a}\right) \]

       2. associate-*r* 82.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot 1\right) \cdot \frac{y \cdot x}{a}} \]

       3. metadata-eval 82.7%

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

       4. associate-*r/ 82.7%

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

       5. associate-*r* 82.7%

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

       6. /-rgt-identity 82.7%

          \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\frac{y}{1}}\right) \cdot x}{a} \]

       7. *-commutative 82.7%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.5 \cdot \frac{y}{1}\right)}}{a} \]

       8. associate-*l/ 84.5%

          \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(0.5 \cdot \frac{y}{1}\right)} \]

       9. /-rgt-identity 84.5%

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

       10. *-commutative 84.5%

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

   12. Simplified 84.5%

       \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \]

Alternative 6: 70.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

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) -2e-44)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 2e-133)
     (/ (* t -4.5) (/ a z))
     (if (<= (* x y) 5e+59)
       (* (* x y) (/ 0.5 a))
       (if (<= (* x y) 1e+130)
         (* -4.5 (* z (/ t a)))
         (* (* y 0.5) (/ x a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = (t * -4.5) / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Fortran

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) <= (-2d-44)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 2d-133) then
        tmp = (t * (-4.5d0)) / (a / z)
    else if ((x * y) <= 5d+59) then
        tmp = (x * y) * (0.5d0 / a)
    else if ((x * y) <= 1d+130) then
        tmp = (-4.5d0) * (z * (t / a))
    else
        tmp = (y * 0.5d0) * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = (t * -4.5) / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = (x * y) * (0.5 / a);
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-44:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 2e-133:
		tmp = (t * -4.5) / (a / z)
	elif (x * y) <= 5e+59:
		tmp = (x * y) * (0.5 / a)
	elif (x * y) <= 1e+130:
		tmp = -4.5 * (z * (t / a))
	else:
		tmp = (y * 0.5) * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-44)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 2e-133)
		tmp = Float64(Float64(t * -4.5) / Float64(a / z));
	elseif (Float64(x * y) <= 5e+59)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	elseif (Float64(x * y) <= 1e+130)
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	else
		tmp = Float64(Float64(y * 0.5) * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-44)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 2e-133)
		tmp = (t * -4.5) / (a / z);
	elseif ((x * y) <= 5e+59)
		tmp = (x * y) * (0.5 / a);
	elseif ((x * y) <= 1e+130)
		tmp = -4.5 * (z * (t / a));
	else
		tmp = (y * 0.5) * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

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], -2e-44], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-133], N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+59], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999991e-44

   1. Initial program 90.4%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.4%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.4%

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

      5. fma-neg 91.8%

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

      6. associate-*l* 91.8%

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

      7. distribute-rgt-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. distribute-rgt-neg-in 91.8%

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

      10. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

   6. Simplified 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r/ 69.9%

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

      3. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -1.99999999999999991e-44 < (*.f64 x y) < 2.0000000000000001e-133

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. +-rgt-identity 90.3%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.3%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.3%

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

      5. fma-neg 90.3%

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

      6. associate-*l* 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

      1. associate-*r/ 84.1%

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

   8. Applied egg-rr 84.1%

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

   if 2.0000000000000001e-133 < (*.f64 x y) < 4.9999999999999997e59

   1. Initial program 96.9%

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

      1. sub-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. neg-sub0 96.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]

      4. associate-+l- 96.9%

         \[\leadsto \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]

      5. sub0-neg 96.9%

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

      6. neg-mul-1 96.9%

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

      7. associate-/l* 96.1%

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

      8. associate-/r/ 96.8%

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

      9. *-commutative 96.8%

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

      10. sub-neg 96.8%

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

      11. +-commutative 96.8%

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

      12. neg-sub0 96.8%

          \[\leadsto \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]

      13. associate-+l- 96.8%

          \[\leadsto \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]

      14. sub0-neg 96.8%

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

      15. distribute-lft-neg-out 96.8%

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

      16. distribute-rgt-neg-in 96.8%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around inf 65.3%

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

   if 4.9999999999999997e59 < (*.f64 x y) < 1.0000000000000001e130

   1. Initial program 88.8%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 88.8%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 88.8%

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

      5. fma-neg 88.8%

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

      6. associate-*l* 89.2%

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

      7. distribute-rgt-neg-in 89.2%

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

      8. *-commutative 89.2%

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

      9. distribute-rgt-neg-in 89.2%

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

      10. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 64.3%

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

      1. associate-/l* 63.9%

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

      2. associate-/r/ 74.5%

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

   6. Simplified 74.5%

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

   if 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 92.0%

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

      1. sub-neg 92.0%

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

      2. +-commutative 92.0%

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

      3. associate-*l* 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. fma-def 94.0%

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

      6. *-commutative 94.0%

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

      7. distribute-rgt-neg-in 94.0%

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

      8. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

      1. add-cube-cbrt 93.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}} \]

      2. pow3 93.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right)}^{3}} \]

      3. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot 2}}}\right)}^{3} \]

      4. metadata-eval 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}}}\right)}^{3} \]

      5. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}}}\right)}^{3} \]

      6. clear-num 93.5%

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

   5. Applied egg-rr 93.5%

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

      1. cbrt-prod 93.3%

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

   7. Applied egg-rr 93.3%

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

      1. *-commutative 93.3%

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

   9. Simplified 93.3%

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

   10. Taylor expanded in z around 0 82.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{y \cdot x}{a}\right)} \]
   11. Step-by-step derivation

       1. pow-base-1 82.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{1} \cdot \frac{y \cdot x}{a}\right) \]

       2. associate-*r* 82.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot 1\right) \cdot \frac{y \cdot x}{a}} \]

       3. metadata-eval 82.7%

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

       4. associate-*r/ 82.7%

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

       5. associate-*r* 82.7%

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

       6. /-rgt-identity 82.7%

          \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\frac{y}{1}}\right) \cdot x}{a} \]

       7. *-commutative 82.7%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.5 \cdot \frac{y}{1}\right)}}{a} \]

       8. associate-*l/ 84.5%

          \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(0.5 \cdot \frac{y}{1}\right)} \]

       9. /-rgt-identity 84.5%

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

       10. *-commutative 84.5%

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

   12. Simplified 84.5%

       \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \]

Alternative 7: 70.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

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) -2e-44)
   (* x (* y (/ 0.5 a)))
   (if (<= (* x y) 2e-133)
     (/ (* t -4.5) (/ a z))
     (if (<= (* x y) 5e+59)
       (/ (* (* x y) 0.5) a)
       (if (<= (* x y) 1e+130)
         (* -4.5 (* z (/ t a)))
         (* (* y 0.5) (/ x a)))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = (t * -4.5) / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = ((x * y) * 0.5) / a;
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Fortran

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) <= (-2d-44)) then
        tmp = x * (y * (0.5d0 / a))
    else if ((x * y) <= 2d-133) then
        tmp = (t * (-4.5d0)) / (a / z)
    else if ((x * y) <= 5d+59) then
        tmp = ((x * y) * 0.5d0) / a
    else if ((x * y) <= 1d+130) then
        tmp = (-4.5d0) * (z * (t / a))
    else
        tmp = (y * 0.5d0) * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-44) {
		tmp = x * (y * (0.5 / a));
	} else if ((x * y) <= 2e-133) {
		tmp = (t * -4.5) / (a / z);
	} else if ((x * y) <= 5e+59) {
		tmp = ((x * y) * 0.5) / a;
	} else if ((x * y) <= 1e+130) {
		tmp = -4.5 * (z * (t / a));
	} else {
		tmp = (y * 0.5) * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-44:
		tmp = x * (y * (0.5 / a))
	elif (x * y) <= 2e-133:
		tmp = (t * -4.5) / (a / z)
	elif (x * y) <= 5e+59:
		tmp = ((x * y) * 0.5) / a
	elif (x * y) <= 1e+130:
		tmp = -4.5 * (z * (t / a))
	else:
		tmp = (y * 0.5) * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-44)
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	elseif (Float64(x * y) <= 2e-133)
		tmp = Float64(Float64(t * -4.5) / Float64(a / z));
	elseif (Float64(x * y) <= 5e+59)
		tmp = Float64(Float64(Float64(x * y) * 0.5) / a);
	elseif (Float64(x * y) <= 1e+130)
		tmp = Float64(-4.5 * Float64(z * Float64(t / a)));
	else
		tmp = Float64(Float64(y * 0.5) * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-44)
		tmp = x * (y * (0.5 / a));
	elseif ((x * y) <= 2e-133)
		tmp = (t * -4.5) / (a / z);
	elseif ((x * y) <= 5e+59)
		tmp = ((x * y) * 0.5) / a;
	elseif ((x * y) <= 1e+130)
		tmp = -4.5 * (z * (t / a));
	else
		tmp = (y * 0.5) * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

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], -2e-44], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-133], N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+59], N[(N[(N[(x * y), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], N[(-4.5 * N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] * N[(x / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999991e-44

   1. Initial program 90.4%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.4%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.4%

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

      5. fma-neg 91.8%

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

      6. associate-*l* 91.8%

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

      7. distribute-rgt-neg-in 91.8%

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

      8. *-commutative 91.8%

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

      9. distribute-rgt-neg-in 91.8%

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

      10. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

   6. Simplified 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r/ 69.9%

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

      3. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -1.99999999999999991e-44 < (*.f64 x y) < 2.0000000000000001e-133

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. +-rgt-identity 90.3%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 90.3%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 90.3%

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

      5. fma-neg 90.3%

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

      6. associate-*l* 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 79.1%

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

      1. associate-/l* 84.1%

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

   6. Simplified 84.1%

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

      1. associate-*r/ 84.1%

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

   8. Applied egg-rr 84.1%

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

   if 2.0000000000000001e-133 < (*.f64 x y) < 4.9999999999999997e59

   1. Initial program 96.9%

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

      1. div-sub 96.9%

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

      2. +-rgt-identity 96.9%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 96.9%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 96.9%

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

      5. fma-neg 96.9%

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

      6. associate-*l* 96.9%

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

      7. distribute-rgt-neg-in 96.9%

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

      8. *-commutative 96.9%

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

      9. distribute-rgt-neg-in 96.9%

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

      10. metadata-eval 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. *-commutative 65.4%

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

      3. *-commutative 65.4%

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

      4. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   if 4.9999999999999997e59 < (*.f64 x y) < 1.0000000000000001e130

   1. Initial program 88.8%

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

      1. div-sub 89.0%

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

      2. +-rgt-identity 89.0%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 88.8%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 88.8%

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

      5. fma-neg 88.8%

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

      6. associate-*l* 89.2%

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

      7. distribute-rgt-neg-in 89.2%

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

      8. *-commutative 89.2%

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

      9. distribute-rgt-neg-in 89.2%

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

      10. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 64.3%

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

      1. associate-/l* 63.9%

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

      2. associate-/r/ 74.5%

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

   6. Simplified 74.5%

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

   if 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 92.0%

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

      1. sub-neg 92.0%

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

      2. +-commutative 92.0%

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

      3. associate-*l* 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. fma-def 94.0%

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

      6. *-commutative 94.0%

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

      7. distribute-rgt-neg-in 94.0%

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

      8. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

      1. add-cube-cbrt 93.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}} \]

      2. pow3 93.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right)}^{3}} \]

      3. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot 2}}}\right)}^{3} \]

      4. metadata-eval 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}}}\right)}^{3} \]

      5. div-inv 93.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}}}\right)}^{3} \]

      6. clear-num 93.5%

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

   5. Applied egg-rr 93.5%

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

      1. cbrt-prod 93.3%

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

   7. Applied egg-rr 93.3%

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

      1. *-commutative 93.3%

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

   9. Simplified 93.3%

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

   10. Taylor expanded in z around 0 82.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{y \cdot x}{a}\right)} \]
   11. Step-by-step derivation

       1. pow-base-1 82.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{1} \cdot \frac{y \cdot x}{a}\right) \]

       2. associate-*r* 82.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot 1\right) \cdot \frac{y \cdot x}{a}} \]

       3. metadata-eval 82.7%

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

       4. associate-*r/ 82.7%

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

       5. associate-*r* 82.7%

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

       6. /-rgt-identity 82.7%

          \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\frac{y}{1}}\right) \cdot x}{a} \]

       7. *-commutative 82.7%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.5 \cdot \frac{y}{1}\right)}}{a} \]

       8. associate-*l/ 84.5%

          \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(0.5 \cdot \frac{y}{1}\right)} \]

       9. /-rgt-identity 84.5%

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

       10. *-commutative 84.5%

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

   12. Simplified 84.5%

       \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(y \cdot 0.5\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.5\right) \cdot \frac{x}{a}\\ \end{array} \]

Alternative 8: 92.9% accurate, 0.8× speedup

Math

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

FPCore

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))
   (/ (* y 0.5) (/ a x))
   (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))))

C

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

Java

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 = (y * 0.5) / (a / x);
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[(N[(y * 0.5), $MachinePrecision] / N[(a / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.8%

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

      1. sub-neg 48.8%

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

      2. +-commutative 48.8%

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

      3. associate-*l* 48.8%

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

      4. distribute-rgt-neg-in 48.8%

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

      5. fma-def 48.8%

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

      6. *-commutative 48.8%

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

      7. distribute-rgt-neg-in 48.8%

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

      8. metadata-eval 48.8%

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

   3. Simplified 48.8%

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

      1. add-cube-cbrt 48.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}} \]

      2. pow3 48.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}}\right)}^{3}} \]

      3. div-inv 48.8%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot 2}}}\right)}^{3} \]

      4. metadata-eval 48.8%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}}}\right)}^{3} \]

      5. div-inv 48.8%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}}}\right)}^{3} \]

      6. clear-num 48.8%

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

   5. Applied egg-rr 48.8%

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

      1. cbrt-prod 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. *-commutative 48.8%

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

   9. Simplified 48.8%

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

   10. Taylor expanded in z around 0 48.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{y \cdot x}{a}\right)} \]
   11. Step-by-step derivation

       1. pow-base-1 48.8%

          \[\leadsto 0.5 \cdot \left(\color{blue}{1} \cdot \frac{y \cdot x}{a}\right) \]

       2. associate-*r* 48.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot 1\right) \cdot \frac{y \cdot x}{a}} \]

       3. metadata-eval 48.8%

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

       4. associate-*r/ 48.8%

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

       5. associate-*r* 48.8%

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

       6. /-rgt-identity 48.8%

          \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\frac{y}{1}}\right) \cdot x}{a} \]

       7. *-commutative 48.8%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.5 \cdot \frac{y}{1}\right)}}{a} \]

       8. associate-*l/ 84.8%

          \[\leadsto \color{blue}{\frac{x}{a} \cdot \left(0.5 \cdot \frac{y}{1}\right)} \]

       9. /-rgt-identity 84.8%

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

       10. *-commutative 84.8%

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

   12. Simplified 84.8%

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

       1. *-commutative 84.8%

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

       2. clear-num 84.9%

          \[\leadsto \left(y \cdot 0.5\right) \cdot \color{blue}{\frac{1}{\frac{a}{x}}} \]

       3. un-div-inv 84.9%

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

   14. Applied egg-rr 84.9%

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

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

   1. Initial program 93.8%

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

      1. associate-*l* 93.8%

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

   3. Simplified 93.8%

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

4. Final simplification 93.4%

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

Alternative 9: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -150000000 \lor \neg \left(z \leq 1.75 \cdot 10^{-131}\right):\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -150000000.0) (not (<= z 1.75e-131)))
   (* -4.5 (/ t (/ a z)))
   (* x (* y (/ 0.5 a)))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -150000000.0) || !(z <= 1.75e-131)) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = x * (y * (0.5 / a));
	}
	return tmp;
}

Fortran

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 ((z <= (-150000000.0d0)) .or. (.not. (z <= 1.75d-131))) then
        tmp = (-4.5d0) * (t / (a / z))
    else
        tmp = x * (y * (0.5d0 / a))
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -150000000.0) || !(z <= 1.75e-131)) {
		tmp = -4.5 * (t / (a / z));
	} else {
		tmp = x * (y * (0.5 / a));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z <= -150000000.0) or not (z <= 1.75e-131):
		tmp = -4.5 * (t / (a / z))
	else:
		tmp = x * (y * (0.5 / a))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -150000000.0) || !(z <= 1.75e-131))
		tmp = Float64(-4.5 * Float64(t / Float64(a / z)));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 / a)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -150000000.0) || ~((z <= 1.75e-131)))
		tmp = -4.5 * (t / (a / z));
	else
		tmp = x * (y * (0.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -150000000.0], N[Not[LessEqual[z, 1.75e-131]], $MachinePrecision]], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e8 or 1.7500000000000001e-131 < z

   1. Initial program 89.1%

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

      1. div-sub 87.9%

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

      2. +-rgt-identity 87.9%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 89.1%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 89.1%

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

      5. fma-neg 90.3%

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

      6. associate-*l* 90.3%

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

      7. distribute-rgt-neg-in 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt-neg-in 90.3%

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

      10. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 51.2%

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

      1. associate-/l* 55.7%

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

   6. Simplified 55.7%

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

   if -1.5e8 < z < 1.7500000000000001e-131

   1. Initial program 95.6%

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

      1. div-sub 95.6%

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

      2. +-rgt-identity 95.6%

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      3. div-sub 95.6%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      4. +-rgt-identity 95.6%

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

      5. fma-neg 95.6%

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

      6. associate-*l* 95.6%

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

      7. distribute-rgt-neg-in 95.6%

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

      8. *-commutative 95.6%

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

      9. distribute-rgt-neg-in 95.6%

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

      10. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around inf 73.3%

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

      1. associate-*r/ 73.3%

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

      2. *-commutative 73.3%

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

      3. *-commutative 73.3%

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

      4. *-commutative 73.3%

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

   6. Simplified 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-*r/ 73.4%

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

      3. associate-*l* 70.9%

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

   8. Applied egg-rr 70.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -150000000 \lor \neg \left(z \leq 1.75 \cdot 10^{-131}\right):\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \end{array} \]

Alternative 10: 51.0% accurate, 1.9× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 91.5%

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

   1. div-sub 90.7%

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

   2. +-rgt-identity 90.7%

      \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   3. div-sub 91.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

   4. +-rgt-identity 91.5%

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

   5. fma-neg 92.3%

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

   6. associate-*l* 92.3%

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

   7. distribute-rgt-neg-in 92.3%

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

   8. *-commutative 92.3%

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

   9. distribute-rgt-neg-in 92.3%

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

   10. metadata-eval 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in x around 0 46.5%

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

   1. associate-/l* 49.0%

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

   2. associate-/r/ 48.3%

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

6. Simplified 48.3%

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

7. Final simplification 48.3%

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

Alternative 11: 50.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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) * (t / (a / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. div-sub 90.7%

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

   2. +-rgt-identity 90.7%

      \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   3. div-sub 91.5%

      \[\leadsto \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

   4. +-rgt-identity 91.5%

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

   5. fma-neg 92.3%

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

   6. associate-*l* 92.3%

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

   7. distribute-rgt-neg-in 92.3%

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

   8. *-commutative 92.3%

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

   9. distribute-rgt-neg-in 92.3%

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

   10. metadata-eval 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in x around 0 46.5%

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

   1. associate-/l* 49.0%

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

6. Simplified 49.0%

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

7. Final simplification 49.0%

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

Developer target: 93.2% 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 2023240 
(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)))