Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 85.6% → 89.1%

Time: 10.4s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+264}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e+264)
   (- (/ y a) (/ (/ x a) z))
   (if (<= z 1.4e+53) (/ (- x (* z y)) (- t (* z a))) (/ y (- a (/ t z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+264) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.4e+53) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	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 (z <= (-3.6d+264)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 1.4d+53) then
        tmp = (x - (z * y)) / (t - (z * a))
    else
        tmp = y / (a - (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+264) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.4e+53) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e+264:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 1.4e+53:
		tmp = (x - (z * y)) / (t - (z * a))
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e+264)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 1.4e+53)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.6e+264)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 1.4e+53)
		tmp = (x - (z * y)) / (t - (z * a));
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e+264], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e+53], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.60000000000000011e264

   1. Initial program 28.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 28.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 28.4%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

         \[\leadsto \frac{y}{a} + \frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{\color{blue}{z}} \]

      8. distribute-lft-out-- N/A

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

      9. associate-*r/ N/A

         \[\leadsto \frac{y}{a} + -1 \cdot \color{blue}{\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{a}\right), \color{blue}{\left(-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \left(\color{blue}{-1} \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \left(\frac{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}{\color{blue}{z}}\right)\right) \]

      13. distribute-lft-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \left(\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}\right)\right) \]

   7. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{y}{a} + \frac{\frac{y \cdot t}{a \cdot a} - \frac{x}{a}}{z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{a}\right)}, z\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x}{a}\right)\right), z\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{/.f64}\left(\left(0 - \frac{x}{a}\right), z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x}{a}\right)\right), z\right)\right) \]

      4. /-lowering-/.f64 91.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(x, a\right)\right), z\right)\right) \]

   10. Simplified 91.0%

       \[\leadsto \frac{y}{a} + \frac{\color{blue}{0 - \frac{x}{a}}}{z} \]

   if -3.60000000000000011e264 < z < 1.4e53

   1. Initial program 97.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if 1.4e53 < z

   1. Initial program 58.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 58.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{\left(z \cdot \left(\frac{t}{z} - a\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{z} - a\right)}\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{t}{z}\right), \color{blue}{a}\right)\right)\right) \]

      3. /-lowering-/.f64 58.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(t, z\right), a\right)\right)\right) \]

   7. Simplified 58.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{\frac{t}{z} - a}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{y}{\frac{t}{z} - a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(\left(\frac{t}{z} - a\right)\right)}} \]

      3. mul-1-neg N/A

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

      4. distribute-lft-out-- N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} - \color{blue}{-1 \cdot a}} \]

      5. sub-neg N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right)} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + a} \]

      8. +-commutative N/A

         \[\leadsto \frac{y}{a + \color{blue}{-1 \cdot \frac{t}{z}}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a + -1 \cdot \frac{t}{z}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a - \color{blue}{\frac{t}{z}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \color{blue}{\left(\frac{t}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 86.1%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 86.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+264}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.56 \cdot 10^{-12}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-83}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 0.235:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.56e-12)
   (/ (- y (/ x z)) a)
   (if (<= z 1.45e-83)
     (/ (- x (* z y)) t)
     (if (<= z 0.235) (/ x (- t (* z a))) (/ y (- a (/ t z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.56e-12) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.45e-83) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 0.235) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	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 (z <= (-1.56d-12)) then
        tmp = (y - (x / z)) / a
    else if (z <= 1.45d-83) then
        tmp = (x - (z * y)) / t
    else if (z <= 0.235d0) then
        tmp = x / (t - (z * a))
    else
        tmp = y / (a - (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.56e-12) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.45e-83) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 0.235) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.56e-12:
		tmp = (y - (x / z)) / a
	elif z <= 1.45e-83:
		tmp = (x - (z * y)) / t
	elif z <= 0.235:
		tmp = x / (t - (z * a))
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.56e-12)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1.45e-83)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 0.235)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.56e-12)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1.45e-83)
		tmp = (x - (z * y)) / t;
	elseif (z <= 0.235)
		tmp = x / (t - (z * a));
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.56e-12], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.45e-83], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 0.235], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.56000000000000002e-12

   1. Initial program 77.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \frac{x - y \cdot z}{z}}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x - y \cdot z}{z}\right), \color{blue}{a}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - \frac{y \cdot z}{z}\right)\right), a\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - y \cdot \frac{z}{z}\right)\right), a\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - y \cdot 1\right)\right), a\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - y\right)\right), a\right) \]

      10. distribute-lft-out-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} - -1 \cdot y\right), a\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right), a\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), a\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + y\right), a\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + -1 \cdot \frac{x}{z}\right), a\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right), a\right) \]

      16. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{x}{z}\right), a\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{x}{z}\right)\right), a\right) \]

      18. /-lowering-/.f64 77.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(x, z\right)\right), a\right) \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a}} \]

   if -1.56000000000000002e-12 < z < 1.45e-83

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 85.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   7. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]

   if 1.45e-83 < z < 0.23499999999999999

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 83.5%

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

   if 0.23499999999999999 < z

   1. Initial program 64.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{\left(z \cdot \left(\frac{t}{z} - a\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{z} - a\right)}\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{t}{z}\right), \color{blue}{a}\right)\right)\right) \]

      3. /-lowering-/.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(t, z\right), a\right)\right)\right) \]

   7. Simplified 64.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{\frac{t}{z} - a}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{y}{\frac{t}{z} - a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(\left(\frac{t}{z} - a\right)\right)}} \]

      3. mul-1-neg N/A

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

      4. distribute-lft-out-- N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} - \color{blue}{-1 \cdot a}} \]

      5. sub-neg N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right)} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + a} \]

      8. +-commutative N/A

         \[\leadsto \frac{y}{a + \color{blue}{-1 \cdot \frac{t}{z}}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a + -1 \cdot \frac{t}{z}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a - \color{blue}{\frac{t}{z}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \color{blue}{\left(\frac{t}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 80.8%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 80.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.56 \cdot 10^{-12}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-83}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 0.235:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 5.6:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -1.35e+32)
     t_1
     (if (<= z 1.6e-83)
       (/ (- x (* z y)) t)
       (if (<= z 5.6) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -1.35e+32) {
		tmp = t_1;
	} else if (z <= 1.6e-83) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 5.6) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	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 = y / (a - (t / z))
    if (z <= (-1.35d+32)) then
        tmp = t_1
    else if (z <= 1.6d-83) then
        tmp = (x - (z * y)) / t
    else if (z <= 5.6d0) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -1.35e+32) {
		tmp = t_1;
	} else if (z <= 1.6e-83) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 5.6) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -1.35e+32:
		tmp = t_1
	elif z <= 1.6e-83:
		tmp = (x - (z * y)) / t
	elif z <= 5.6:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -1.35e+32)
		tmp = t_1;
	elseif (z <= 1.6e-83)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 5.6)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -1.35e+32)
		tmp = t_1;
	elseif (z <= 1.6e-83)
		tmp = (x - (z * y)) / t;
	elseif (z <= 5.6)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.35e+32], t$95$1, If[LessEqual[z, 1.6e-83], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 5.6], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35000000000000006e32 or 5.5999999999999996 < z

   1. Initial program 68.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{\left(z \cdot \left(\frac{t}{z} - a\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{z} - a\right)}\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{t}{z}\right), \color{blue}{a}\right)\right)\right) \]

      3. /-lowering-/.f64 68.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(t, z\right), a\right)\right)\right) \]

   7. Simplified 68.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{\frac{t}{z} - a}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{y}{\frac{t}{z} - a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(\left(\frac{t}{z} - a\right)\right)}} \]

      3. mul-1-neg N/A

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

      4. distribute-lft-out-- N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} - \color{blue}{-1 \cdot a}} \]

      5. sub-neg N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right)} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + a} \]

      8. +-commutative N/A

         \[\leadsto \frac{y}{a + \color{blue}{-1 \cdot \frac{t}{z}}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a + -1 \cdot \frac{t}{z}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a - \color{blue}{\frac{t}{z}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \color{blue}{\left(\frac{t}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 78.6%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 78.6%

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

   if -1.35000000000000006e32 < z < 1.6000000000000001e-83

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   7. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]

   if 1.6000000000000001e-83 < z < 5.5999999999999996

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 83.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 5.6:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{0 - y}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e+46)
   (/ y a)
   (if (<= z -3.4e-66)
     (* z (/ (- 0.0 y) t))
     (if (<= z 1.65e+14) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+46) {
		tmp = y / a;
	} else if (z <= -3.4e-66) {
		tmp = z * ((0.0 - y) / t);
	} else if (z <= 1.65e+14) {
		tmp = x / t;
	} else {
		tmp = y / 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 (z <= (-7.2d+46)) then
        tmp = y / a
    else if (z <= (-3.4d-66)) then
        tmp = z * ((0.0d0 - y) / t)
    else if (z <= 1.65d+14) then
        tmp = x / t
    else
        tmp = y / 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 (z <= -7.2e+46) {
		tmp = y / a;
	} else if (z <= -3.4e-66) {
		tmp = z * ((0.0 - y) / t);
	} else if (z <= 1.65e+14) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e+46:
		tmp = y / a
	elif z <= -3.4e-66:
		tmp = z * ((0.0 - y) / t)
	elif z <= 1.65e+14:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e+46)
		tmp = Float64(y / a);
	elseif (z <= -3.4e-66)
		tmp = Float64(z * Float64(Float64(0.0 - y) / t));
	elseif (z <= 1.65e+14)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e+46)
		tmp = y / a;
	elseif (z <= -3.4e-66)
		tmp = z * ((0.0 - y) / t);
	elseif (z <= 1.65e+14)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e+46], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.4e-66], N[(z * N[(N[(0.0 - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.65e+14], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.1999999999999997e46 or 1.65e14 < z

   1. Initial program 67.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -7.1999999999999997e46 < z < -3.39999999999999997e-66

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   7. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{y}{t} + \frac{x}{t \cdot z}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(-1 \cdot \frac{y}{t} + \frac{x}{t \cdot z}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{x}{t \cdot z} + \color{blue}{-1 \cdot \frac{y}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{x}{t \cdot z} + \left(\mathsf{neg}\left(\frac{y}{t}\right)\right)\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\frac{x}{t \cdot z} - \color{blue}{\frac{y}{t}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{x}{t \cdot z}\right), \color{blue}{\left(\frac{y}{t}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \left(t \cdot z\right)\right), \left(\frac{\color{blue}{y}}{t}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, z\right)\right), \left(\frac{y}{t}\right)\right)\right) \]

      8. /-lowering-/.f64 67.4%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(t, z\right)\right), \mathsf{/.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   10. Simplified 67.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(-1 \cdot \frac{y}{t}\right)}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\frac{y}{t}\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \left(0 - \color{blue}{\frac{y}{t}}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y}{t}\right)}\right)\right) \]

       4. /-lowering-/.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(y, \color{blue}{t}\right)\right)\right) \]

   13. Simplified 60.6%

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

   if -3.39999999999999997e-66 < z < 1.65e14

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{t}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

   7. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{x}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{0 - y}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.8:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -2.4e-65) t_1 (if (<= z 0.8) (/ x (- t (* z a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -2.4e-65) {
		tmp = t_1;
	} else if (z <= 0.8) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	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 = y / (a - (t / z))
    if (z <= (-2.4d-65)) then
        tmp = t_1
    else if (z <= 0.8d0) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -2.4e-65) {
		tmp = t_1;
	} else if (z <= 0.8) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -2.4e-65:
		tmp = t_1
	elif z <= 0.8:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -2.4e-65)
		tmp = t_1;
	elseif (z <= 0.8)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -2.4e-65)
		tmp = t_1;
	elseif (z <= 0.8)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e-65], t$95$1, If[LessEqual[z, 0.8], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000002e-65 or 0.80000000000000004 < z

   1. Initial program 71.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{\left(z \cdot \left(\frac{t}{z} - a\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{t}{z} - a\right)}\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{t}{z}\right), \color{blue}{a}\right)\right)\right) \]

      3. /-lowering-/.f64 71.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(t, z\right), a\right)\right)\right) \]

   7. Simplified 71.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{\frac{t}{z} - a}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{y}{\frac{t}{z} - a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(\left(\frac{t}{z} - a\right)\right)}} \]

      3. mul-1-neg N/A

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

      4. distribute-lft-out-- N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} - \color{blue}{-1 \cdot a}} \]

      5. sub-neg N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right)} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{-1 \cdot \frac{t}{z} + a} \]

      8. +-commutative N/A

         \[\leadsto \frac{y}{a + \color{blue}{-1 \cdot \frac{t}{z}}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(a + -1 \cdot \frac{t}{z}\right)}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a + \left(\mathsf{neg}\left(\frac{t}{z}\right)\right)\right)\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(a - \color{blue}{\frac{t}{z}}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \color{blue}{\left(\frac{t}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, \mathsf{/.f64}\left(t, \color{blue}{z}\right)\right)\right) \]

   10. Simplified 77.8%

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

   if -2.4000000000000002e-65 < z < 0.80000000000000004

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 77.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 77.5%

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

Alternative 6: 66.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.9e+42) (/ y a) (if (<= z 5.4e+45) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.9e+42) {
		tmp = y / a;
	} else if (z <= 5.4e+45) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / 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 (z <= (-4.9d+42)) then
        tmp = y / a
    else if (z <= 5.4d+45) then
        tmp = x / (t - (z * a))
    else
        tmp = y / 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 (z <= -4.9e+42) {
		tmp = y / a;
	} else if (z <= 5.4e+45) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.9e+42:
		tmp = y / a
	elif z <= 5.4e+45:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.9e+42)
		tmp = Float64(y / a);
	elseif (z <= 5.4e+45)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.9e+42)
		tmp = y / a;
	elseif (z <= 5.4e+45)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.9e+42], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.4e+45], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9000000000000002e42 or 5.39999999999999968e45 < z

   1. Initial program 65.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 64.4%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -4.9000000000000002e42 < z < 5.39999999999999968e45

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 73.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 73.1%

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

Alternative 7: 56.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 60000000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+29) (/ y a) (if (<= z 60000000000000.0) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+29) {
		tmp = y / a;
	} else if (z <= 60000000000000.0) {
		tmp = x / t;
	} else {
		tmp = y / 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 (z <= (-1.3d+29)) then
        tmp = y / a
    else if (z <= 60000000000000.0d0) then
        tmp = x / t
    else
        tmp = y / 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 (z <= -1.3e+29) {
		tmp = y / a;
	} else if (z <= 60000000000000.0) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.3e+29:
		tmp = y / a
	elif z <= 60000000000000.0:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+29)
		tmp = Float64(y / a);
	elseif (z <= 60000000000000.0)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.3e+29)
		tmp = y / a;
	elseif (z <= 60000000000000.0)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+29], N[(y / a), $MachinePrecision], If[LessEqual[z, 60000000000000.0], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e29 or 6e13 < z

   1. Initial program 68.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 62.0%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   7. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -1.3e29 < z < 6e13

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{t}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 58.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{t} \end{array} \]

FPCore

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

C

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

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 / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x / t), $MachinePrecision]

Derivation

1. Initial program 84.7%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{\left(t - a \cdot z\right)}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), \left(\color{blue}{t} - a \cdot z\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(t - a \cdot z\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

   6. *-lowering-*.f64 84.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

3. Simplified 84.7%

   \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 36.0%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

7. Simplified 36.0%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
8. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / 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 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / 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 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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