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

Percentage Accurate: 84.7% → 90.9%

Time: 15.4s

Alternatives: 9

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: 84.7% 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: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.3e+179)
   (/ y (- a (/ t z)))
   (if (<= z 1.55e+109) (/ (- x (* z y)) (- t (* z a))) (/ (- y (/ x z)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.3e+179) {
		tmp = y / (a - (t / z));
	} else if (z <= 1.55e+109) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / 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) :: tmp
    if (z <= (-3.3d+179)) then
        tmp = y / (a - (t / z))
    else if (z <= 1.55d+109) then
        tmp = (x - (z * y)) / (t - (z * a))
    else
        tmp = (y - (x / z)) / 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 <= -3.3e+179) {
		tmp = y / (a - (t / z));
	} else if (z <= 1.55e+109) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.3e+179:
		tmp = y / (a - (t / z))
	elif z <= 1.55e+109:
		tmp = (x - (z * y)) / (t - (z * a))
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.3e+179)
		tmp = Float64(y / Float64(a - Float64(t / z)));
	elseif (z <= 1.55e+109)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.3e+179)
		tmp = y / (a - (t / z));
	elseif (z <= 1.55e+109)
		tmp = (x - (z * y)) / (t - (z * a));
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.3e+179], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e+109], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.29999999999999978e179

   1. Initial program 51.1%

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

      1. *-commutative 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in z around inf 51.1%

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

   6. Taylor expanded in x around 0 95.5%

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

      1. mul-1-neg 95.5%

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

      2. distribute-neg-frac2 95.5%

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

      3. sub-neg 95.5%

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

      4. +-commutative 95.5%

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

      5. distribute-neg-in 95.5%

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

      6. remove-double-neg 95.5%

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

      7. unsub-neg 95.5%

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

   8. Simplified 95.5%

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

   if -3.29999999999999978e179 < z < 1.54999999999999996e109

   1. Initial program 95.0%

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

   if 1.54999999999999996e109 < z

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in y around -inf 57.2%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in a around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. associate-*r* 83.7%

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

      3. neg-mul-1 83.7%

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

      4. sub-neg 83.7%

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

      5. associate-/r* 72.9%

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

      6. metadata-eval 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in y around 0 86.4%

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

       1. mul-1-neg 86.4%

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

       2. unsub-neg 86.4%

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

   13. Simplified 86.4%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+179)
   (/ y a)
   (if (<= z -6.4e+147)
     (/ (/ x z) (- a))
     (if (<= z -4e+25)
       (/ y a)
       (if (<= z -3.9e-54)
         (/ (* z y) (- t))
         (if (<= z 7.4e-93) (/ x t) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+179) {
		tmp = y / a;
	} else if (z <= -6.4e+147) {
		tmp = (x / z) / -a;
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -3.9e-54) {
		tmp = (z * y) / -t;
	} else if (z <= 7.4e-93) {
		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 <= (-3.2d+179)) then
        tmp = y / a
    else if (z <= (-6.4d+147)) then
        tmp = (x / z) / -a
    else if (z <= (-4d+25)) then
        tmp = y / a
    else if (z <= (-3.9d-54)) then
        tmp = (z * y) / -t
    else if (z <= 7.4d-93) 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 <= -3.2e+179) {
		tmp = y / a;
	} else if (z <= -6.4e+147) {
		tmp = (x / z) / -a;
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -3.9e-54) {
		tmp = (z * y) / -t;
	} else if (z <= 7.4e-93) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+179:
		tmp = y / a
	elif z <= -6.4e+147:
		tmp = (x / z) / -a
	elif z <= -4e+25:
		tmp = y / a
	elif z <= -3.9e-54:
		tmp = (z * y) / -t
	elif z <= 7.4e-93:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+179)
		tmp = Float64(y / a);
	elseif (z <= -6.4e+147)
		tmp = Float64(Float64(x / z) / Float64(-a));
	elseif (z <= -4e+25)
		tmp = Float64(y / a);
	elseif (z <= -3.9e-54)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= 7.4e-93)
		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 <= -3.2e+179)
		tmp = y / a;
	elseif (z <= -6.4e+147)
		tmp = (x / z) / -a;
	elseif (z <= -4e+25)
		tmp = y / a;
	elseif (z <= -3.9e-54)
		tmp = (z * y) / -t;
	elseif (z <= 7.4e-93)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+179], N[(y / a), $MachinePrecision], If[LessEqual[z, -6.4e+147], N[(N[(x / z), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[z, -4e+25], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.9e-54], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, 7.4e-93], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.2000000000000002e179 or -6.39999999999999958e147 < z < -4.00000000000000036e25 or 7.40000000000000005e-93 < z

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in z around inf 58.6%

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

   if -3.2000000000000002e179 < z < -6.39999999999999958e147

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in y around -inf 71.2%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in a around inf 32.3%

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

      1. associate-*r/ 32.3%

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

      2. associate-*r* 32.3%

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

      3. neg-mul-1 32.3%

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

      4. sub-neg 32.3%

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

      5. associate-/r* 32.2%

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

      6. metadata-eval 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in y around 0 51.8%

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

       1. associate-*r/ 51.8%

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

       2. mul-1-neg 51.8%

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

   13. Simplified 51.8%

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

   if -4.00000000000000036e25 < z < -3.9e-54

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. associate-/l* 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

      4. distribute-neg-frac2 61.4%

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

      5. cancel-sign-sub-inv 61.4%

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

      6. *-commutative 61.4%

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

      7. +-commutative 61.4%

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

      8. *-commutative 61.4%

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

      9. neg-mul-1 61.4%

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

      10. associate-*r* 61.4%

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

      11. mul-1-neg 61.4%

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

      12. distribute-rgt-neg-in 61.4%

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

      13. fma-undefine 61.4%

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

      14. neg-sub0 61.4%

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

      15. fma-undefine 61.4%

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

      16. distribute-rgt-neg-in 61.4%

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

      17. mul-1-neg 61.4%

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

      18. associate-*r* 61.4%

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

      19. neg-mul-1 61.4%

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

      20. *-commutative 61.4%

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

      21. associate--r+ 61.4%

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

   7. Simplified 61.4%

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

   8. Taylor expanded in z around 0 54.9%

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

      1. associate-*r/ 54.9%

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

      2. neg-mul-1 54.9%

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

   10. Simplified 54.9%

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

   11. Taylor expanded in y around 0 54.9%

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

       1. associate-*r/ 54.9%

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

       2. *-commutative 54.9%

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

       3. neg-mul-1 54.9%

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

       4. distribute-rgt-neg-in 54.9%

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

   13. Simplified 54.9%

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

   if -3.9e-54 < z < 7.40000000000000005e-93

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 57.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+117}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-54}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e+179)
   (/ y a)
   (if (<= z -5e+117)
     (* y (/ z (- t)))
     (if (<= z -4e+25)
       (/ y a)
       (if (<= z -1.65e-54)
         (/ (* z y) (- t))
         (if (<= z 7.4e-93) (/ x t) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+179) {
		tmp = y / a;
	} else if (z <= -5e+117) {
		tmp = y * (z / -t);
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -1.65e-54) {
		tmp = (z * y) / -t;
	} else if (z <= 7.4e-93) {
		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 <= (-3.6d+179)) then
        tmp = y / a
    else if (z <= (-5d+117)) then
        tmp = y * (z / -t)
    else if (z <= (-4d+25)) then
        tmp = y / a
    else if (z <= (-1.65d-54)) then
        tmp = (z * y) / -t
    else if (z <= 7.4d-93) 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 <= -3.6e+179) {
		tmp = y / a;
	} else if (z <= -5e+117) {
		tmp = y * (z / -t);
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -1.65e-54) {
		tmp = (z * y) / -t;
	} else if (z <= 7.4e-93) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.6e+179:
		tmp = y / a
	elif z <= -5e+117:
		tmp = y * (z / -t)
	elif z <= -4e+25:
		tmp = y / a
	elif z <= -1.65e-54:
		tmp = (z * y) / -t
	elif z <= 7.4e-93:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e+179)
		tmp = Float64(y / a);
	elseif (z <= -5e+117)
		tmp = Float64(y * Float64(z / Float64(-t)));
	elseif (z <= -4e+25)
		tmp = Float64(y / a);
	elseif (z <= -1.65e-54)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= 7.4e-93)
		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 <= -3.6e+179)
		tmp = y / a;
	elseif (z <= -5e+117)
		tmp = y * (z / -t);
	elseif (z <= -4e+25)
		tmp = y / a;
	elseif (z <= -1.65e-54)
		tmp = (z * y) / -t;
	elseif (z <= 7.4e-93)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e+179], N[(y / a), $MachinePrecision], If[LessEqual[z, -5e+117], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4e+25], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.65e-54], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, 7.4e-93], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.5999999999999998e179 or -4.99999999999999983e117 < z < -4.00000000000000036e25 or 7.40000000000000005e-93 < z

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

   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 59.7%

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

   if -3.5999999999999998e179 < z < -4.99999999999999983e117

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 30.9%

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

      1. mul-1-neg 30.9%

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

      2. associate-/l* 50.8%

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

      3. distribute-rgt-neg-in 50.8%

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

      4. distribute-neg-frac2 50.8%

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

      5. cancel-sign-sub-inv 50.8%

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

      6. *-commutative 50.8%

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

      7. +-commutative 50.8%

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

      8. *-commutative 50.8%

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

      9. neg-mul-1 50.8%

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

      10. associate-*r* 50.8%

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

      11. mul-1-neg 50.8%

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

      12. distribute-rgt-neg-in 50.8%

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

      13. fma-undefine 50.8%

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

      14. neg-sub0 50.8%

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

      15. fma-undefine 50.8%

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

      16. distribute-rgt-neg-in 50.8%

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

      17. mul-1-neg 50.8%

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

      18. associate-*r* 50.8%

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

      19. neg-mul-1 50.8%

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

      20. *-commutative 50.8%

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

      21. associate--r+ 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in z around 0 43.4%

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

      1. associate-*r/ 43.4%

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

      2. neg-mul-1 43.4%

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

   10. Simplified 43.4%

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

   if -4.00000000000000036e25 < z < -1.64999999999999996e-54

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. associate-/l* 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

      4. distribute-neg-frac2 61.4%

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

      5. cancel-sign-sub-inv 61.4%

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

      6. *-commutative 61.4%

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

      7. +-commutative 61.4%

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

      8. *-commutative 61.4%

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

      9. neg-mul-1 61.4%

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

      10. associate-*r* 61.4%

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

      11. mul-1-neg 61.4%

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

      12. distribute-rgt-neg-in 61.4%

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

      13. fma-undefine 61.4%

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

      14. neg-sub0 61.4%

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

      15. fma-undefine 61.4%

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

      16. distribute-rgt-neg-in 61.4%

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

      17. mul-1-neg 61.4%

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

      18. associate-*r* 61.4%

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

      19. neg-mul-1 61.4%

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

      20. *-commutative 61.4%

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

      21. associate--r+ 61.4%

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

   7. Simplified 61.4%

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

   8. Taylor expanded in z around 0 54.9%

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

      1. associate-*r/ 54.9%

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

      2. neg-mul-1 54.9%

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

   10. Simplified 54.9%

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

   11. Taylor expanded in y around 0 54.9%

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

       1. associate-*r/ 54.9%

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

       2. *-commutative 54.9%

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

       3. neg-mul-1 54.9%

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

       4. distribute-rgt-neg-in 54.9%

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

   13. Simplified 54.9%

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

   if -1.64999999999999996e-54 < z < 7.40000000000000005e-93

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 57.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+117}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-54}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{-t}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z (- t)))))
   (if (<= z -3.6e+179)
     (/ y a)
     (if (<= z -1.9e+119)
       t_1
       (if (<= z -4e+25)
         (/ y a)
         (if (<= z -1.05e-54) t_1 (if (<= z 5.5e-93) (/ x t) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / -t);
	double tmp;
	if (z <= -3.6e+179) {
		tmp = y / a;
	} else if (z <= -1.9e+119) {
		tmp = t_1;
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -1.05e-54) {
		tmp = t_1;
	} else if (z <= 5.5e-93) {
		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) :: t_1
    real(8) :: tmp
    t_1 = y * (z / -t)
    if (z <= (-3.6d+179)) then
        tmp = y / a
    else if (z <= (-1.9d+119)) then
        tmp = t_1
    else if (z <= (-4d+25)) then
        tmp = y / a
    else if (z <= (-1.05d-54)) then
        tmp = t_1
    else if (z <= 5.5d-93) 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 t_1 = y * (z / -t);
	double tmp;
	if (z <= -3.6e+179) {
		tmp = y / a;
	} else if (z <= -1.9e+119) {
		tmp = t_1;
	} else if (z <= -4e+25) {
		tmp = y / a;
	} else if (z <= -1.05e-54) {
		tmp = t_1;
	} else if (z <= 5.5e-93) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / -t)
	tmp = 0
	if z <= -3.6e+179:
		tmp = y / a
	elif z <= -1.9e+119:
		tmp = t_1
	elif z <= -4e+25:
		tmp = y / a
	elif z <= -1.05e-54:
		tmp = t_1
	elif z <= 5.5e-93:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / Float64(-t)))
	tmp = 0.0
	if (z <= -3.6e+179)
		tmp = Float64(y / a);
	elseif (z <= -1.9e+119)
		tmp = t_1;
	elseif (z <= -4e+25)
		tmp = Float64(y / a);
	elseif (z <= -1.05e-54)
		tmp = t_1;
	elseif (z <= 5.5e-93)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / -t);
	tmp = 0.0;
	if (z <= -3.6e+179)
		tmp = y / a;
	elseif (z <= -1.9e+119)
		tmp = t_1;
	elseif (z <= -4e+25)
		tmp = y / a;
	elseif (z <= -1.05e-54)
		tmp = t_1;
	elseif (z <= 5.5e-93)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e+179], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.9e+119], t$95$1, If[LessEqual[z, -4e+25], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.05e-54], t$95$1, If[LessEqual[z, 5.5e-93], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999998e179 or -1.89999999999999995e119 < z < -4.00000000000000036e25 or 5.49999999999999968e-93 < z

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

   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 59.7%

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

   if -3.5999999999999998e179 < z < -1.89999999999999995e119 or -4.00000000000000036e25 < z < -1.05e-54

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. associate-/l* 58.0%

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

      3. distribute-rgt-neg-in 58.0%

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

      4. distribute-neg-frac2 58.0%

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

      5. cancel-sign-sub-inv 58.0%

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

      6. *-commutative 58.0%

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

      7. +-commutative 58.0%

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

      8. *-commutative 58.0%

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

      9. neg-mul-1 58.0%

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

      10. associate-*r* 58.0%

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

      11. mul-1-neg 58.0%

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

      12. distribute-rgt-neg-in 58.0%

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

      13. fma-undefine 58.0%

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

      14. neg-sub0 58.0%

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

      15. fma-undefine 58.0%

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

      16. distribute-rgt-neg-in 58.0%

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

      17. mul-1-neg 58.0%

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

      18. associate-*r* 58.0%

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

      19. neg-mul-1 58.0%

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

      20. *-commutative 58.0%

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

      21. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in z around 0 51.3%

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

      1. associate-*r/ 51.3%

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

      2. neg-mul-1 51.3%

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

   10. Simplified 51.3%

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

   if -1.05e-54 < z < 5.49999999999999968e-93

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 57.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6500000 \lor \neg \left(a \leq -2.9 \cdot 10^{-33}\right) \land \left(a \leq -4.2 \cdot 10^{-77} \lor \neg \left(a \leq 1.45 \cdot 10^{-105}\right)\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6500000.0)
         (and (not (<= a -2.9e-33))
              (or (<= a -4.2e-77) (not (<= a 1.45e-105)))))
   (/ (- y (/ x z)) a)
   (/ (- x (* z y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6500000.0) || (!(a <= -2.9e-33) && ((a <= -4.2e-77) || !(a <= 1.45e-105)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	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 <= (-6500000.0d0)) .or. (.not. (a <= (-2.9d-33))) .and. (a <= (-4.2d-77)) .or. (.not. (a <= 1.45d-105))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (z * y)) / t
    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 <= -6500000.0) || (!(a <= -2.9e-33) && ((a <= -4.2e-77) || !(a <= 1.45e-105)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6500000.0) or (not (a <= -2.9e-33) and ((a <= -4.2e-77) or not (a <= 1.45e-105))):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (z * y)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6500000.0) || (!(a <= -2.9e-33) && ((a <= -4.2e-77) || !(a <= 1.45e-105))))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6500000.0) || (~((a <= -2.9e-33)) && ((a <= -4.2e-77) || ~((a <= 1.45e-105)))))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (z * y)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6500000.0], And[N[Not[LessEqual[a, -2.9e-33]], $MachinePrecision], Or[LessEqual[a, -4.2e-77], N[Not[LessEqual[a, 1.45e-105]], $MachinePrecision]]]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5e6 or -2.90000000000000003e-33 < a < -4.20000000000000031e-77 or 1.45000000000000002e-105 < a

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in y around -inf 71.4%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in a around inf 74.7%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.7%

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

      3. neg-mul-1 74.7%

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

      4. sub-neg 74.7%

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

      5. associate-/r* 72.0%

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

      6. metadata-eval 72.0%

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

   10. Simplified 72.0%

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

   11. Taylor expanded in y around 0 82.7%

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

       1. mul-1-neg 82.7%

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

       2. unsub-neg 82.7%

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

   13. Simplified 82.7%

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

   if -6.5e6 < a < -2.90000000000000003e-33 or -4.20000000000000031e-77 < a < 1.45000000000000002e-105

   1. Initial program 96.4%

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

      1. *-commutative 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t around inf 89.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6500000 \lor \neg \left(a \leq -2.9 \cdot 10^{-33}\right) \land \left(a \leq -4.2 \cdot 10^{-77} \lor \neg \left(a \leq 1.45 \cdot 10^{-105}\right)\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= y -2.9e+103)
     (/ y a)
     (if (<= y 4.2e+21)
       t_1
       (if (<= y 2.4e+53)
         (/ y a)
         (if (<= y 2.8e+171) t_1 (* y (/ z (- t)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (y <= -2.9e+103) {
		tmp = y / a;
	} else if (y <= 4.2e+21) {
		tmp = t_1;
	} else if (y <= 2.4e+53) {
		tmp = y / a;
	} else if (y <= 2.8e+171) {
		tmp = t_1;
	} else {
		tmp = y * (z / -t);
	}
	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 = x / (t - (z * a))
    if (y <= (-2.9d+103)) then
        tmp = y / a
    else if (y <= 4.2d+21) then
        tmp = t_1
    else if (y <= 2.4d+53) then
        tmp = y / a
    else if (y <= 2.8d+171) then
        tmp = t_1
    else
        tmp = y * (z / -t)
    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 / (t - (z * a));
	double tmp;
	if (y <= -2.9e+103) {
		tmp = y / a;
	} else if (y <= 4.2e+21) {
		tmp = t_1;
	} else if (y <= 2.4e+53) {
		tmp = y / a;
	} else if (y <= 2.8e+171) {
		tmp = t_1;
	} else {
		tmp = y * (z / -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if y <= -2.9e+103:
		tmp = y / a
	elif y <= 4.2e+21:
		tmp = t_1
	elif y <= 2.4e+53:
		tmp = y / a
	elif y <= 2.8e+171:
		tmp = t_1
	else:
		tmp = y * (z / -t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (y <= -2.9e+103)
		tmp = Float64(y / a);
	elseif (y <= 4.2e+21)
		tmp = t_1;
	elseif (y <= 2.4e+53)
		tmp = Float64(y / a);
	elseif (y <= 2.8e+171)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(z / Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (y <= -2.9e+103)
		tmp = y / a;
	elseif (y <= 4.2e+21)
		tmp = t_1;
	elseif (y <= 2.4e+53)
		tmp = y / a;
	elseif (y <= 2.8e+171)
		tmp = t_1;
	else
		tmp = y * (z / -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.9e+103], N[(y / a), $MachinePrecision], If[LessEqual[y, 4.2e+21], t$95$1, If[LessEqual[y, 2.4e+53], N[(y / a), $MachinePrecision], If[LessEqual[y, 2.8e+171], t$95$1, N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8999999999999998e103 or 4.2e21 < y < 2.4e53

   1. Initial program 69.0%

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

      1. *-commutative 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in z around inf 64.6%

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

   if -2.8999999999999998e103 < y < 4.2e21 or 2.4e53 < y < 2.80000000000000004e171

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 71.2%

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

      1. *-commutative 71.2%

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

   7. Simplified 71.2%

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

   if 2.80000000000000004e171 < y

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in x around 0 49.1%

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

      1. mul-1-neg 49.1%

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

      2. associate-/l* 58.0%

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

      3. distribute-rgt-neg-in 58.0%

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

      4. distribute-neg-frac2 58.0%

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

      5. cancel-sign-sub-inv 58.0%

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

      6. *-commutative 58.0%

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

      7. +-commutative 58.0%

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

      8. *-commutative 58.0%

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

      9. neg-mul-1 58.0%

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

      10. associate-*r* 58.0%

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

      11. mul-1-neg 58.0%

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

      12. distribute-rgt-neg-in 58.0%

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

      13. fma-undefine 58.0%

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

      14. neg-sub0 58.0%

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

      15. fma-undefine 58.0%

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

      16. distribute-rgt-neg-in 58.0%

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

      17. mul-1-neg 58.0%

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

      18. associate-*r* 58.0%

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

      19. neg-mul-1 58.0%

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

      20. *-commutative 58.0%

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

      21. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in z around 0 49.3%

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

      1. associate-*r/ 49.3%

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

      2. neg-mul-1 49.3%

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

   10. Simplified 49.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+171}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-25} \lor \neg \left(x \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -1.65e-25) (not (<= x 2e-19)))
   (/ x (- t (* z a)))
   (/ y (- a (/ t z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -1.65e-25) || !(x <= 2e-19)) {
		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 ((x <= (-1.65d-25)) .or. (.not. (x <= 2d-19))) 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 ((x <= -1.65e-25) || !(x <= 2e-19)) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x <= -1.65e-25) or not (x <= 2e-19):
		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 ((x <= -1.65e-25) || !(x <= 2e-19))
		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 ((x <= -1.65e-25) || ~((x <= 2e-19)))
		tmp = x / (t - (z * a));
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -1.65e-25], N[Not[LessEqual[x, 2e-19]], $MachinePrecision]], 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 2 regimes

2. if x < -1.6499999999999999e-25 or 2e-19 < x

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if -1.6499999999999999e-25 < x < 2e-19

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in z around inf 77.2%

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

   6. Taylor expanded in x around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. distribute-neg-frac2 78.6%

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

      3. sub-neg 78.6%

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

      4. +-commutative 78.6%

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

      5. distribute-neg-in 78.6%

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

      6. remove-double-neg 78.6%

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

      7. unsub-neg 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-25} \lor \neg \left(x \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -43000000 \lor \neg \left(z \leq 7.4 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -43000000.0) (not (<= z 7.4e-93))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -43000000.0) || !(z <= 7.4e-93)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	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 <= (-43000000.0d0)) .or. (.not. (z <= 7.4d-93))) then
        tmp = y / a
    else
        tmp = x / t
    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 <= -43000000.0) || !(z <= 7.4e-93)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -43000000.0) or not (z <= 7.4e-93):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -43000000.0) || !(z <= 7.4e-93))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -43000000.0) || ~((z <= 7.4e-93)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -43000000.0], N[Not[LessEqual[z, 7.4e-93]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3e7 or 7.40000000000000005e-93 < z

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in z around inf 53.2%

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

   if -4.3e7 < z < 7.40000000000000005e-93

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   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 52.0%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -43000000 \lor \neg \left(z \leq 7.4 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% 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 83.6%

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

   1. *-commutative 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in z around 0 32.7%

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

Developer target: 97.3% 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 2024110 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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