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

Percentage Accurate: 90.9% → 93.0%

Time: 9.4s

Alternatives: 14

Speedup: 0.6×

• 26.9% 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: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+302}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) -2e+302)
   (* z (/ (* t -4.5) a))
   (/ (fma x y (* z (* t -9.0))) (* a 2.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= -2e+302) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = fma(x, y, (z * (t * -9.0))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= -2e+302)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], -2e+302], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $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 z 9) t) < -2.0000000000000002e302

   1. Initial program 57.0%

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

   3. Taylor expanded in x around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. associate-*r* 67.1%

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

      3. associate-*l/ 100.0%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   if -2.0000000000000002e302 < (*.f64 (*.f64 z 9) t)

   1. Initial program 94.6%

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

      1. div-sub 93.3%

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

      2. *-commutative 93.3%

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

      3. div-sub 94.6%

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

      4. cancel-sign-sub-inv 94.6%

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

      5. *-commutative 94.6%

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

      6. fma-define 95.0%

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

      7. distribute-rgt-neg-in 95.0%

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

      8. associate-*r* 95.0%

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

      9. distribute-lft-neg-in 95.0%

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

      10. *-commutative 95.0%

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

      11. distribute-rgt-neg-in 95.0%

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

      12. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+302}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+210}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) -5e+210)
   (* z (/ (* t -4.5) a))
   (* (fma x y (* t (* z -9.0))) (/ 0.5 a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= -5e+210) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = fma(x, y, (t * (z * -9.0))) * (0.5 / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= -5e+210)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	else
		tmp = Float64(fma(x, y, Float64(t * Float64(z * -9.0))) * Float64(0.5 / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], -5e+210], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z 9) t) < -4.9999999999999998e210

   1. Initial program 73.1%

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

   3. Taylor expanded in x around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. associate-*r* 79.4%

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

      3. associate-*l/ 99.9%

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

      4. associate-*r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.9%

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

   5. Simplified 99.9%

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

   if -4.9999999999999998e210 < (*.f64 (*.f64 z 9) t)

   1. Initial program 94.3%

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

      1. div-inv 94.2%

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

      2. fma-neg 94.7%

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

      3. *-commutative 94.7%

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

      4. distribute-rgt-neg-in 94.7%

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

      5. distribute-rgt-neg-in 94.7%

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

      6. metadata-eval 94.7%

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

      7. *-commutative 94.7%

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

      8. associate-/r* 94.7%

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

      9. metadata-eval 94.7%

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

   4. Applied egg-rr 94.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+210}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (<= (* x y) -2e-87)
       (* (/ 0.5 a) (/ y (/ 1.0 x)))
       (if (<= (* x y) 5e-99)
         (/ (* z (* t -9.0)) (* a 2.0))
         (/ (* x y) (* a 2.0)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (y / (1.0 / x));
	} else if ((x * y) <= 5e-99) {
		tmp = (z * (t * -9.0)) / (a * 2.0);
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if ((x * y) <= (-2d-87)) then
        tmp = (0.5d0 / a) * (y / (1.0d0 / x))
    else if ((x * y) <= 5d-99) then
        tmp = (z * (t * (-9.0d0))) / (a * 2.0d0)
    else
        tmp = (x * y) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (y / (1.0 / x));
	} else if ((x * y) <= 5e-99) {
		tmp = (z * (t * -9.0)) / (a * 2.0);
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif (x * y) <= -2e-87:
		tmp = (0.5 / a) * (y / (1.0 / x))
	elif (x * y) <= 5e-99:
		tmp = (z * (t * -9.0)) / (a * 2.0)
	else:
		tmp = (x * y) / (a * 2.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif (Float64(x * y) <= -2e-87)
		tmp = Float64(Float64(0.5 / a) * Float64(y / Float64(1.0 / x)));
	elseif (Float64(x * y) <= 5e-99)
		tmp = Float64(Float64(z * Float64(t * -9.0)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif ((x * y) <= -2e-87)
		tmp = (0.5 / a) * (y / (1.0 / x));
	elseif ((x * y) <= 5e-99)
		tmp = (z * (t * -9.0)) / (a * 2.0);
	else
		tmp = (x * y) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[(N[(0.5 / a), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-99], N[(N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 56.2%

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

      1. fma-define 56.2%

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

      2. *-commutative 56.2%

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

      3. associate-/r* 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-/l* 72.9%

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

      6. associate-*r/ 72.9%

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

   5. Simplified 72.9%

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

   6. Taylor expanded in z around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. *-commutative 51.7%

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

      2. div-inv 51.7%

         \[\leadsto \frac{0.5 \cdot y}{\color{blue}{a \cdot \frac{1}{x}}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}} \]

   10. Applied egg-rr 77.4%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}} \]

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*r* 83.5%

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

   5. Simplified 83.5%

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

   if 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (or (<= (* x y) -2e-87) (not (<= (* x y) 5e-99)))
       (* (/ 0.5 a) (* x y))
       (* -4.5 (/ (* z t) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if (((x * y) <= (-2d-87)) .or. (.not. ((x * y) <= 5d-99))) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = (-4.5d0) * ((z * t) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = -4.5 * ((z * t) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif ((x * y) <= -2e-87) or not ((x * y) <= 5e-99):
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = -4.5 * ((z * t) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif ((Float64(x * y) <= -2e-87) || !(Float64(x * y) <= 5e-99))
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif (((x * y) <= -2e-87) || ~(((x * y) <= 5e-99)))
		tmp = (0.5 / a) * (x * y);
	else
		tmp = -4.5 * ((z * t) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-99]], $MachinePrecision]], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87 or 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 91.0%

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

      1. div-inv 90.9%

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

      2. fma-neg 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. distribute-rgt-neg-in 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 92.0%

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

      9. metadata-eval 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot \left(z \cdot t\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (or (<= (* x y) -2e-87) (not (<= (* x y) 5e-99)))
       (* (/ 0.5 a) (* x y))
       (/ (* -4.5 (* z t)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = (-4.5 * (z * t)) / a;
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if (((x * y) <= (-2d-87)) .or. (.not. ((x * y) <= 5d-99))) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = ((-4.5d0) * (z * t)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = (-4.5 * (z * t)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif ((x * y) <= -2e-87) or not ((x * y) <= 5e-99):
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = (-4.5 * (z * t)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif ((Float64(x * y) <= -2e-87) || !(Float64(x * y) <= 5e-99))
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = Float64(Float64(-4.5 * Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif (((x * y) <= -2e-87) || ~(((x * y) <= 5e-99)))
		tmp = (0.5 / a) * (x * y);
	else
		tmp = (-4.5 * (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-99]], $MachinePrecision]], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(-4.5 * N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87 or 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 91.0%

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

      1. div-inv 90.9%

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

      2. fma-neg 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. distribute-rgt-neg-in 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 92.0%

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

      9. metadata-eval 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

      1. associate-*r/ 83.5%

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

   5. Applied egg-rr 83.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot \left(z \cdot t\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (or (<= (* x y) -2e-87) (not (<= (* x y) 5e-99)))
       (* (/ 0.5 a) (* x y))
       (/ (* t (* z -4.5)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = (t * (z * -4.5)) / a;
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if (((x * y) <= (-2d-87)) .or. (.not. ((x * y) <= 5d-99))) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = (t * (z * (-4.5d0))) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = (t * (z * -4.5)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif ((x * y) <= -2e-87) or not ((x * y) <= 5e-99):
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = (t * (z * -4.5)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif ((Float64(x * y) <= -2e-87) || !(Float64(x * y) <= 5e-99))
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = Float64(Float64(t * Float64(z * -4.5)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif (((x * y) <= -2e-87) || ~(((x * y) <= 5e-99)))
		tmp = (0.5 / a) * (x * y);
	else
		tmp = (t * (z * -4.5)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-99]], $MachinePrecision]], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87 or 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 91.0%

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

      1. div-inv 90.9%

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

      2. fma-neg 92.0%

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

      3. *-commutative 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. distribute-rgt-neg-in 92.0%

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

      6. metadata-eval 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/r* 92.0%

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

      9. metadata-eval 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

      1. associate-*r/ 83.5%

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

      2. associate-*r* 83.5%

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

      3. associate-*l/ 73.5%

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

      4. associate-*r/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l* 73.4%

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

   5. Simplified 73.4%

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

      1. associate-*l/ 83.5%

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

      2. *-commutative 83.5%

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

   7. Applied egg-rr 83.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (<= (* x y) -2e-87)
       (* (/ 0.5 a) (* x y))
       (if (<= (* x y) 5e-99) (/ (* t (* z -4.5)) a) (/ (* x y) (* a 2.0)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (x * y);
	} else if ((x * y) <= 5e-99) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if ((x * y) <= (-2d-87)) then
        tmp = (0.5d0 / a) * (x * y)
    else if ((x * y) <= 5d-99) then
        tmp = (t * (z * (-4.5d0))) / a
    else
        tmp = (x * y) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (x * y);
	} else if ((x * y) <= 5e-99) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif (x * y) <= -2e-87:
		tmp = (0.5 / a) * (x * y)
	elif (x * y) <= 5e-99:
		tmp = (t * (z * -4.5)) / a
	else:
		tmp = (x * y) / (a * 2.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif (Float64(x * y) <= -2e-87)
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	elseif (Float64(x * y) <= 5e-99)
		tmp = Float64(Float64(t * Float64(z * -4.5)) / a);
	else
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif ((x * y) <= -2e-87)
		tmp = (0.5 / a) * (x * y);
	elseif ((x * y) <= 5e-99)
		tmp = (t * (z * -4.5)) / a;
	else
		tmp = (x * y) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-99], N[(N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87

   1. Initial program 99.6%

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

      1. div-inv 99.4%

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

      2. fma-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-/r* 99.4%

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

      9. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around inf 77.3%

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

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

      1. associate-*r/ 83.5%

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

      2. associate-*r* 83.5%

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

      3. associate-*l/ 73.5%

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

      4. associate-*r/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l* 73.4%

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

   5. Simplified 73.4%

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

      1. associate-*l/ 83.5%

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

      2. *-commutative 83.5%

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

   7. Applied egg-rr 83.5%

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

   if 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+39)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -1e-46)
     (* z (/ (* t -4.5) a))
     (if (<= (* x y) -2e-87)
       (* (/ 0.5 a) (/ y (/ 1.0 x)))
       (if (<= (* x y) 5e-99) (/ (* t (* z -4.5)) a) (/ (* x y) (* a 2.0)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (y / (1.0 / x));
	} else if ((x * y) <= 5e-99) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	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 ((x * y) <= (-2d+39)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else if ((x * y) <= (-2d-87)) then
        tmp = (0.5d0 / a) * (y / (1.0d0 / x))
    else if ((x * y) <= 5d-99) then
        tmp = (t * (z * (-4.5d0))) / a
    else
        tmp = (x * y) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -1e-46) {
		tmp = z * ((t * -4.5) / a);
	} else if ((x * y) <= -2e-87) {
		tmp = (0.5 / a) * (y / (1.0 / x));
	} else if ((x * y) <= 5e-99) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = (x * y) / (a * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -1e-46:
		tmp = z * ((t * -4.5) / a)
	elif (x * y) <= -2e-87:
		tmp = (0.5 / a) * (y / (1.0 / x))
	elif (x * y) <= 5e-99:
		tmp = (t * (z * -4.5)) / a
	else:
		tmp = (x * y) / (a * 2.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	elseif (Float64(x * y) <= -2e-87)
		tmp = Float64(Float64(0.5 / a) * Float64(y / Float64(1.0 / x)));
	elseif (Float64(x * y) <= 5e-99)
		tmp = Float64(Float64(t * Float64(z * -4.5)) / a);
	else
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -1e-46)
		tmp = z * ((t * -4.5) / a);
	elseif ((x * y) <= -2e-87)
		tmp = (0.5 / a) * (y / (1.0 / x));
	elseif ((x * y) <= 5e-99)
		tmp = (t * (z * -4.5)) / a;
	else
		tmp = (x * y) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[(N[(0.5 / a), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-99], N[(N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 73.4%

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

      3. associate-*r* 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 74.8%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.8%

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

      3. associate-*l/ 74.7%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 74.7%

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

   5. Simplified 74.7%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 56.2%

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

      1. fma-define 56.2%

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

      2. *-commutative 56.2%

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

      3. associate-/r* 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-/l* 72.9%

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

      6. associate-*r/ 72.9%

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

   5. Simplified 72.9%

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

   6. Taylor expanded in z around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. clear-num 51.4%

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

      3. un-div-inv 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. *-commutative 51.7%

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

      2. div-inv 51.7%

         \[\leadsto \frac{0.5 \cdot y}{\color{blue}{a \cdot \frac{1}{x}}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}} \]

   10. Applied egg-rr 77.4%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}} \]

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 83.4%

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

      1. associate-*r/ 83.5%

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

      2. associate-*r* 83.5%

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

      3. associate-*l/ 73.5%

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

      4. associate-*r/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l* 73.4%

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

   5. Simplified 73.4%

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

      1. associate-*l/ 83.5%

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

      2. *-commutative 83.5%

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

   7. Applied egg-rr 83.5%

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

   if 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 73.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{y}{\frac{1}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ t_2 := -4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a))) (t_2 (* -4.5 (/ (* z t) a))))
   (if (<= z -5.2e+42)
     t_2
     (if (<= z 1.7e-145)
       t_1
       (if (<= z 3.8) t_2 (if (<= z 3.3e+49) t_1 (* -4.5 (* t (/ z a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = -4.5 * ((z * t) / a);
	double tmp;
	if (z <= -5.2e+42) {
		tmp = t_2;
	} else if (z <= 1.7e-145) {
		tmp = t_1;
	} else if (z <= 3.8) {
		tmp = t_2;
	} else if (z <= 3.3e+49) {
		tmp = t_1;
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a)
    t_2 = (-4.5d0) * ((z * t) / a)
    if (z <= (-5.2d+42)) then
        tmp = t_2
    else if (z <= 1.7d-145) then
        tmp = t_1
    else if (z <= 3.8d0) then
        tmp = t_2
    else if (z <= 3.3d+49) then
        tmp = t_1
    else
        tmp = (-4.5d0) * (t * (z / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = -4.5 * ((z * t) / a);
	double tmp;
	if (z <= -5.2e+42) {
		tmp = t_2;
	} else if (z <= 1.7e-145) {
		tmp = t_1;
	} else if (z <= 3.8) {
		tmp = t_2;
	} else if (z <= 3.3e+49) {
		tmp = t_1;
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * ((y * 0.5) / a)
	t_2 = -4.5 * ((z * t) / a)
	tmp = 0
	if z <= -5.2e+42:
		tmp = t_2
	elif z <= 1.7e-145:
		tmp = t_1
	elif z <= 3.8:
		tmp = t_2
	elif z <= 3.3e+49:
		tmp = t_1
	else:
		tmp = -4.5 * (t * (z / a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a))
	t_2 = Float64(-4.5 * Float64(Float64(z * t) / a))
	tmp = 0.0
	if (z <= -5.2e+42)
		tmp = t_2;
	elseif (z <= 1.7e-145)
		tmp = t_1;
	elseif (z <= 3.8)
		tmp = t_2;
	elseif (z <= 3.3e+49)
		tmp = t_1;
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((y * 0.5) / a);
	t_2 = -4.5 * ((z * t) / a);
	tmp = 0.0;
	if (z <= -5.2e+42)
		tmp = t_2;
	elseif (z <= 1.7e-145)
		tmp = t_1;
	elseif (z <= 3.8)
		tmp = t_2;
	elseif (z <= 3.3e+49)
		tmp = t_1;
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+42], t$95$2, If[LessEqual[z, 1.7e-145], t$95$1, If[LessEqual[z, 3.8], t$95$2, If[LessEqual[z, 3.3e+49], t$95$1, N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1999999999999998e42 or 1.6999999999999999e-145 < z < 3.7999999999999998

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 64.7%

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

   if -5.1999999999999998e42 < z < 1.6999999999999999e-145 or 3.7999999999999998 < z < 3.2999999999999998e49

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/l* 64.9%

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

      3. associate-*r* 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-*r/ 64.9%

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

   5. Simplified 64.9%

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

   if 3.2999999999999998e49 < z

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0 56.8%

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

      1. associate-/l* 66.2%

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

   5. Simplified 66.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+42}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;z \leq 3.8:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ t_2 := z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+37}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 13000000000000:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a))) (t_2 (* z (* t (/ -4.5 a)))))
   (if (<= z -2.6e+37)
     t_2
     (if (<= z 1.7e-145)
       t_1
       (if (<= z 13000000000000.0)
         (* -4.5 (/ (* z t) a))
         (if (<= z 2.7e+51) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = z * (t * (-4.5 / a));
	double tmp;
	if (z <= -2.6e+37) {
		tmp = t_2;
	} else if (z <= 1.7e-145) {
		tmp = t_1;
	} else if (z <= 13000000000000.0) {
		tmp = -4.5 * ((z * t) / a);
	} else if (z <= 2.7e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a)
    t_2 = z * (t * ((-4.5d0) / a))
    if (z <= (-2.6d+37)) then
        tmp = t_2
    else if (z <= 1.7d-145) then
        tmp = t_1
    else if (z <= 13000000000000.0d0) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (z <= 2.7d+51) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = z * (t * (-4.5 / a));
	double tmp;
	if (z <= -2.6e+37) {
		tmp = t_2;
	} else if (z <= 1.7e-145) {
		tmp = t_1;
	} else if (z <= 13000000000000.0) {
		tmp = -4.5 * ((z * t) / a);
	} else if (z <= 2.7e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * ((y * 0.5) / a)
	t_2 = z * (t * (-4.5 / a))
	tmp = 0
	if z <= -2.6e+37:
		tmp = t_2
	elif z <= 1.7e-145:
		tmp = t_1
	elif z <= 13000000000000.0:
		tmp = -4.5 * ((z * t) / a)
	elif z <= 2.7e+51:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a))
	t_2 = Float64(z * Float64(t * Float64(-4.5 / a)))
	tmp = 0.0
	if (z <= -2.6e+37)
		tmp = t_2;
	elseif (z <= 1.7e-145)
		tmp = t_1;
	elseif (z <= 13000000000000.0)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif (z <= 2.7e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((y * 0.5) / a);
	t_2 = z * (t * (-4.5 / a));
	tmp = 0.0;
	if (z <= -2.6e+37)
		tmp = t_2;
	elseif (z <= 1.7e-145)
		tmp = t_1;
	elseif (z <= 13000000000000.0)
		tmp = -4.5 * ((z * t) / a);
	elseif (z <= 2.7e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.6e+37], t$95$2, If[LessEqual[z, 1.7e-145], t$95$1, If[LessEqual[z, 13000000000000.0], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+51], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5999999999999999e37 or 2.69999999999999992e51 < z

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0 66.5%

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

      1. associate-*r/ 66.4%

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

   5. Applied egg-rr 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*r/ 66.4%

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

      3. associate-*r* 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-*l* 70.1%

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

   7. Applied egg-rr 70.1%

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

   if -2.5999999999999999e37 < z < 1.6999999999999999e-145 or 1.3e13 < z < 2.69999999999999992e51

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-/l* 65.8%

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

      3. associate-*r* 65.8%

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

      4. *-commutative 65.8%

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

      5. associate-*r/ 65.8%

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

   5. Simplified 65.8%

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

   if 1.6999999999999999e-145 < z < 1.3e13

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0 43.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;z \leq 13000000000000:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ t_2 := z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+37}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-22}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a))) (t_2 (* z (/ (* t -4.5) a))))
   (if (<= z -1.95e+37)
     t_2
     (if (<= z 1.42e-145)
       t_1
       (if (<= z 1.9e-22) (* -4.5 (/ (* z t) a)) (if (<= z 8e+54) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = z * ((t * -4.5) / a);
	double tmp;
	if (z <= -1.95e+37) {
		tmp = t_2;
	} else if (z <= 1.42e-145) {
		tmp = t_1;
	} else if (z <= 1.9e-22) {
		tmp = -4.5 * ((z * t) / a);
	} else if (z <= 8e+54) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a)
    t_2 = z * ((t * (-4.5d0)) / a)
    if (z <= (-1.95d+37)) then
        tmp = t_2
    else if (z <= 1.42d-145) then
        tmp = t_1
    else if (z <= 1.9d-22) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (z <= 8d+54) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double t_2 = z * ((t * -4.5) / a);
	double tmp;
	if (z <= -1.95e+37) {
		tmp = t_2;
	} else if (z <= 1.42e-145) {
		tmp = t_1;
	} else if (z <= 1.9e-22) {
		tmp = -4.5 * ((z * t) / a);
	} else if (z <= 8e+54) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * ((y * 0.5) / a)
	t_2 = z * ((t * -4.5) / a)
	tmp = 0
	if z <= -1.95e+37:
		tmp = t_2
	elif z <= 1.42e-145:
		tmp = t_1
	elif z <= 1.9e-22:
		tmp = -4.5 * ((z * t) / a)
	elif z <= 8e+54:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a))
	t_2 = Float64(z * Float64(Float64(t * -4.5) / a))
	tmp = 0.0
	if (z <= -1.95e+37)
		tmp = t_2;
	elseif (z <= 1.42e-145)
		tmp = t_1;
	elseif (z <= 1.9e-22)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif (z <= 8e+54)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((y * 0.5) / a);
	t_2 = z * ((t * -4.5) / a);
	tmp = 0.0;
	if (z <= -1.95e+37)
		tmp = t_2;
	elseif (z <= 1.42e-145)
		tmp = t_1;
	elseif (z <= 1.9e-22)
		tmp = -4.5 * ((z * t) / a);
	elseif (z <= 8e+54)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e+37], t$95$2, If[LessEqual[z, 1.42e-145], t$95$1, If[LessEqual[z, 1.9e-22], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e+54], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9499999999999999e37 or 8.0000000000000006e54 < z

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0 66.2%

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

      1. associate-*r/ 66.1%

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

      2. associate-*r* 66.2%

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

      3. associate-*l/ 69.9%

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

      4. associate-*r/ 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

   if -1.9499999999999999e37 < z < 1.4200000000000001e-145 or 1.90000000000000012e-22 < z < 8.0000000000000006e54

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-/l* 65.9%

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

      3. associate-*r* 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*r/ 65.9%

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

   5. Simplified 65.9%

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

   if 1.4200000000000001e-145 < z < 1.90000000000000012e-22

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0 43.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-22}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -5e+210) (* z (/ (* t -4.5) a)) (/ (- (* x y) t_1) (* a 2.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -5e+210) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = ((x * y) - t_1) / (a * 2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-5d+210)) then
        tmp = z * ((t * (-4.5d0)) / a)
    else
        tmp = ((x * y) - t_1) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

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 <= -5e+210) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = ((x * y) - t_1) / (a * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -5e+210:
		tmp = z * ((t * -4.5) / a)
	else:
		tmp = ((x * y) - t_1) / (a * 2.0)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -5e+210)
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	else
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -5e+210)
		tmp = z * ((t * -4.5) / a);
	else
		tmp = ((x * y) - t_1) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+210], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z 9) t) < -4.9999999999999998e210

   1. Initial program 73.1%

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

   3. Taylor expanded in x around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. associate-*r* 79.4%

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

      3. associate-*l/ 99.9%

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

      4. associate-*r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.9%

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

   5. Simplified 99.9%

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

   if -4.9999999999999998e210 < (*.f64 (*.f64 z 9) t)

   1. Initial program 94.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+210}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 (* -4.5 (* t (/ z a))))

C

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

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 = (-4.5d0) * (t * (z / a))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -4.5 * (t * (z / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around 0 49.6%

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

   1. associate-/l* 51.0%

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

5. Simplified 51.0%

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

6. Final simplification 51.0%

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

Alternative 14: 51.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -4.5 \cdot \frac{z \cdot t}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* -4.5 (/ (* z t) a)))

C

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

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 = (-4.5d0) * ((z * t) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -4.5 * ((z * t) / a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around 0 49.6%

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

4. Final simplification 49.6%

   \[\leadsto -4.5 \cdot \frac{z \cdot t}{a} \]
5. Add Preprocessing

Developer target: 92.9% accurate, 0.5× 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 2024080 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (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)))