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

Percentage Accurate: 84.8% → 90.3%

Time: 12.4s

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: 84.8% 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: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+90)
   (- (/ y a) (/ (/ x a) z))
   (if (<= z 3.4e+111) (/ (- 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 <= -2.8e+90) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 3.4e+111) {
		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 <= (-2.8d+90)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 3.4d+111) 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 <= -2.8e+90) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 3.4e+111) {
		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 <= -2.8e+90:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 3.4e+111:
		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 <= -2.8e+90)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 3.4e+111)
		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 <= -2.8e+90)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 3.4e+111)
		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, -2.8e+90], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e+111], 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 < -2.8e90

   1. Initial program 53.0%

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

      1. *-commutative 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in y around inf 46.6%

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

   6. Taylor expanded in t around 0 46.5%

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

      1. associate-*r/ 46.5%

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

      2. mul-1-neg 46.5%

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

      3. distribute-rgt-neg-out 46.5%

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

      4. *-commutative 46.5%

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

      5. times-frac 63.9%

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

      6. sub-neg 63.9%

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

      7. +-commutative 63.9%

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

      8. distribute-neg-in 63.9%

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

      9. remove-double-neg 63.9%

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

      10. unsub-neg 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in y around 0 81.3%

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

       1. +-commutative 81.3%

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

       2. mul-1-neg 81.3%

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

       3. unsub-neg 81.3%

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

       4. associate-/r* 90.1%

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

   11. Simplified 90.1%

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

   if -2.8e90 < z < 3.4000000000000001e111

   1. Initial program 98.2%

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

   if 3.4000000000000001e111 < z

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in z around inf 55.4%

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

   6. Taylor expanded in x around 0 84.8%

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

      1. associate-*r/ 84.8%

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

      2. mul-1-neg 84.8%

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

   8. Simplified 84.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-202}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.1e+83)
   (/ y a)
   (if (<= z -7e-36)
     (/ (/ x z) (- a))
     (if (<= z -1.25e-202)
       (/ (* z y) (- t))
       (if (<= z 1.55e+82) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e+83) {
		tmp = y / a;
	} else if (z <= -7e-36) {
		tmp = (x / z) / -a;
	} else if (z <= -1.25e-202) {
		tmp = (z * y) / -t;
	} else if (z <= 1.55e+82) {
		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 <= (-3.1d+83)) then
        tmp = y / a
    else if (z <= (-7d-36)) then
        tmp = (x / z) / -a
    else if (z <= (-1.25d-202)) then
        tmp = (z * y) / -t
    else if (z <= 1.55d+82) 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 <= -3.1e+83) {
		tmp = y / a;
	} else if (z <= -7e-36) {
		tmp = (x / z) / -a;
	} else if (z <= -1.25e-202) {
		tmp = (z * y) / -t;
	} else if (z <= 1.55e+82) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.1e+83:
		tmp = y / a
	elif z <= -7e-36:
		tmp = (x / z) / -a
	elif z <= -1.25e-202:
		tmp = (z * y) / -t
	elif z <= 1.55e+82:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.1e+83)
		tmp = Float64(y / a);
	elseif (z <= -7e-36)
		tmp = Float64(Float64(x / z) / Float64(-a));
	elseif (z <= -1.25e-202)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= 1.55e+82)
		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 <= -3.1e+83)
		tmp = y / a;
	elseif (z <= -7e-36)
		tmp = (x / z) / -a;
	elseif (z <= -1.25e-202)
		tmp = (z * y) / -t;
	elseif (z <= 1.55e+82)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.1e+83], N[(y / a), $MachinePrecision], If[LessEqual[z, -7e-36], N[(N[(x / z), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[z, -1.25e-202], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, 1.55e+82], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.09999999999999992e83 or 1.55000000000000016e82 < z

   1. Initial program 57.4%

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

      1. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in z around inf 70.2%

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

   if -3.09999999999999992e83 < z < -6.9999999999999999e-36

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in z around inf 88.3%

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

   6. Taylor expanded in x around inf 57.5%

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

      1. associate-/r* 65.1%

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

   8. Simplified 65.1%

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

   9. Taylor expanded in t around 0 49.9%

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

       1. neg-mul-1 49.9%

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

   11. Simplified 49.9%

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

   if -6.9999999999999999e-36 < z < -1.24999999999999993e-202

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. associate-/l* 46.9%

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

      3. distribute-rgt-neg-in 46.9%

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

      4. distribute-neg-frac2 46.9%

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

      5. cancel-sign-sub-inv 46.9%

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

      6. *-commutative 46.9%

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

      7. +-commutative 46.9%

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

      8. distribute-rgt-neg-out 46.9%

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

      9. distribute-lft-neg-in 46.9%

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

      10. *-commutative 46.9%

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

      11. fma-undefine 46.9%

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

      12. neg-sub0 46.9%

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

      13. fma-undefine 46.9%

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

      14. distribute-rgt-neg-in 46.9%

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

      15. mul-1-neg 46.9%

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

      16. associate-*r* 46.9%

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

      17. neg-mul-1 46.9%

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

      18. *-commutative 46.9%

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

      19. associate--r+ 46.9%

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

      20. neg-sub0 46.9%

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

      21. distribute-rgt-neg-out 46.9%

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

      22. remove-double-neg 46.9%

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

   7. Simplified 46.9%

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

   8. Taylor expanded in z around 0 49.5%

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

      1. associate-*r/ 49.5%

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

      2. *-commutative 49.5%

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

      3. neg-mul-1 49.5%

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

      4. distribute-lft-neg-in 49.5%

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

   10. Simplified 49.5%

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

   if -1.24999999999999993e-202 < z < 1.55000000000000016e82

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 55.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-202}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{x}{a}}{-z}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e+88)
   (/ y a)
   (if (<= z -1.5e-43)
     (/ (/ x a) (- z))
     (if (<= z -5.1e-203)
       (/ (* z y) (- t))
       (if (<= z 1.35e+82) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e+88) {
		tmp = y / a;
	} else if (z <= -1.5e-43) {
		tmp = (x / a) / -z;
	} else if (z <= -5.1e-203) {
		tmp = (z * y) / -t;
	} else if (z <= 1.35e+82) {
		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 <= (-3.5d+88)) then
        tmp = y / a
    else if (z <= (-1.5d-43)) then
        tmp = (x / a) / -z
    else if (z <= (-5.1d-203)) then
        tmp = (z * y) / -t
    else if (z <= 1.35d+82) 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 <= -3.5e+88) {
		tmp = y / a;
	} else if (z <= -1.5e-43) {
		tmp = (x / a) / -z;
	} else if (z <= -5.1e-203) {
		tmp = (z * y) / -t;
	} else if (z <= 1.35e+82) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.5e+88:
		tmp = y / a
	elif z <= -1.5e-43:
		tmp = (x / a) / -z
	elif z <= -5.1e-203:
		tmp = (z * y) / -t
	elif z <= 1.35e+82:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e+88)
		tmp = Float64(y / a);
	elseif (z <= -1.5e-43)
		tmp = Float64(Float64(x / a) / Float64(-z));
	elseif (z <= -5.1e-203)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= 1.35e+82)
		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 <= -3.5e+88)
		tmp = y / a;
	elseif (z <= -1.5e-43)
		tmp = (x / a) / -z;
	elseif (z <= -5.1e-203)
		tmp = (z * y) / -t;
	elseif (z <= 1.35e+82)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e+88], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.5e-43], N[(N[(x / a), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[z, -5.1e-203], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, 1.35e+82], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.4999999999999998e88 or 1.35e82 < z

   1. Initial program 57.4%

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

      1. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in z around inf 70.2%

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

   if -3.4999999999999998e88 < z < -1.50000000000000002e-43

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in x around inf 57.5%

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

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in t around 0 42.1%

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

      1. mul-1-neg 42.1%

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

      2. associate-/r* 49.7%

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

      3. distribute-neg-frac2 49.7%

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

   10. Simplified 49.7%

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

   if -1.50000000000000002e-43 < z < -5.09999999999999985e-203

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. associate-/l* 46.9%

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

      3. distribute-rgt-neg-in 46.9%

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

      4. distribute-neg-frac2 46.9%

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

      5. cancel-sign-sub-inv 46.9%

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

      6. *-commutative 46.9%

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

      7. +-commutative 46.9%

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

      8. distribute-rgt-neg-out 46.9%

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

      9. distribute-lft-neg-in 46.9%

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

      10. *-commutative 46.9%

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

      11. fma-undefine 46.9%

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

      12. neg-sub0 46.9%

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

      13. fma-undefine 46.9%

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

      14. distribute-rgt-neg-in 46.9%

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

      15. mul-1-neg 46.9%

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

      16. associate-*r* 46.9%

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

      17. neg-mul-1 46.9%

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

      18. *-commutative 46.9%

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

      19. associate--r+ 46.9%

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

      20. neg-sub0 46.9%

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

      21. distribute-rgt-neg-out 46.9%

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

      22. remove-double-neg 46.9%

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

   7. Simplified 46.9%

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

   8. Taylor expanded in z around 0 49.5%

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

      1. associate-*r/ 49.5%

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

      2. *-commutative 49.5%

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

      3. neg-mul-1 49.5%

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

      4. distribute-lft-neg-in 49.5%

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

   10. Simplified 49.5%

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

   if -5.09999999999999985e-203 < z < 1.35e82

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 55.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{x}{a}}{-z}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e-35)
   (- (/ y a) (/ (/ x a) z))
   (if (<= z -1.1e-304)
     (/ (- x (* z y)) t)
     (if (<= z 3.7e+98) (/ 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 <= -1.6e-35) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= -1.1e-304) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.7e+98) {
		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 <= (-1.6d-35)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= (-1.1d-304)) then
        tmp = (x - (z * y)) / t
    else if (z <= 3.7d+98) 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 <= -1.6e-35) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= -1.1e-304) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.7e+98) {
		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 <= -1.6e-35:
		tmp = (y / a) - ((x / a) / z)
	elif z <= -1.1e-304:
		tmp = (x - (z * y)) / t
	elif z <= 3.7e+98:
		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 <= -1.6e-35)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= -1.1e-304)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3.7e+98)
		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 <= -1.6e-35)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= -1.1e-304)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3.7e+98)
		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, -1.6e-35], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.1e-304], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.7e+98], 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 < -1.5999999999999999e-35

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in y around inf 56.7%

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

   6. Taylor expanded in t around 0 46.0%

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

      1. associate-*r/ 46.0%

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

      2. mul-1-neg 46.0%

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

      3. distribute-rgt-neg-out 46.0%

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

      4. *-commutative 46.0%

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

      5. times-frac 56.8%

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

      6. sub-neg 56.8%

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

      7. +-commutative 56.8%

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

      8. distribute-neg-in 56.8%

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

      9. remove-double-neg 56.8%

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

      10. unsub-neg 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in y around 0 70.4%

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

       1. +-commutative 70.4%

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

       2. mul-1-neg 70.4%

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

       3. unsub-neg 70.4%

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

       4. associate-/r* 78.7%

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

   11. Simplified 78.7%

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

   if -1.5999999999999999e-35 < z < -1.1e-304

   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 t around inf 84.0%

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

   if -1.1e-304 < z < 3.6999999999999999e98

   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 inf 76.5%

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

      1. *-commutative 76.5%

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

   7. Simplified 76.5%

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

   if 3.6999999999999999e98 < z

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in z around inf 58.4%

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

   6. Taylor expanded in x around 0 85.8%

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

      1. associate-*r/ 85.8%

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

      2. mul-1-neg 85.8%

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

   8. Simplified 85.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-304}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -5.8e-9)
     t_1
     (if (<= z 2.9e-306)
       (/ (- x (* z y)) t)
       (if (<= z 1.28e+98) (/ 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 <= -5.8e-9) {
		tmp = t_1;
	} else if (z <= 2.9e-306) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.28e+98) {
		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 <= (-5.8d-9)) then
        tmp = t_1
    else if (z <= 2.9d-306) then
        tmp = (x - (z * y)) / t
    else if (z <= 1.28d+98) 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 <= -5.8e-9) {
		tmp = t_1;
	} else if (z <= 2.9e-306) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.28e+98) {
		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 <= -5.8e-9:
		tmp = t_1
	elif z <= 2.9e-306:
		tmp = (x - (z * y)) / t
	elif z <= 1.28e+98:
		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 <= -5.8e-9)
		tmp = t_1;
	elseif (z <= 2.9e-306)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1.28e+98)
		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 <= -5.8e-9)
		tmp = t_1;
	elseif (z <= 2.9e-306)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1.28e+98)
		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, -5.8e-9], t$95$1, If[LessEqual[z, 2.9e-306], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.28e+98], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.79999999999999982e-9 or 1.28000000000000006e98 < z

   1. Initial program 62.4%

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

      1. *-commutative 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in z around inf 62.4%

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

   6. Taylor expanded in x around 0 76.3%

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

      1. associate-*r/ 76.3%

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

      2. mul-1-neg 76.3%

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

   8. Simplified 76.3%

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

   if -5.79999999999999982e-9 < z < 2.8999999999999999e-306

   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 t around inf 83.2%

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

   if 2.8999999999999999e-306 < z < 1.28000000000000006e98

   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 inf 76.5%

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

      1. *-commutative 76.5%

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

   7. Simplified 76.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e+43)
   (/ y a)
   (if (<= z -6.4e-305)
     (/ (- x (* z y)) t)
     (if (<= z 7.2e+99) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+43) {
		tmp = y / a;
	} else if (z <= -6.4e-305) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 7.2e+99) {
		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 <= (-2.6d+43)) then
        tmp = y / a
    else if (z <= (-6.4d-305)) then
        tmp = (x - (z * y)) / t
    else if (z <= 7.2d+99) 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 <= -2.6e+43) {
		tmp = y / a;
	} else if (z <= -6.4e-305) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 7.2e+99) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e+43:
		tmp = y / a
	elif z <= -6.4e-305:
		tmp = (x - (z * y)) / t
	elif z <= 7.2e+99:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e+43)
		tmp = Float64(y / a);
	elseif (z <= -6.4e-305)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 7.2e+99)
		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 <= -2.6e+43)
		tmp = y / a;
	elseif (z <= -6.4e-305)
		tmp = (x - (z * y)) / t;
	elseif (z <= 7.2e+99)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e+43], N[(y / a), $MachinePrecision], If[LessEqual[z, -6.4e-305], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 7.2e+99], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000021e43 or 7.2000000000000003e99 < z

   1. Initial program 58.5%

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

      1. *-commutative 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in z around inf 67.3%

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

   if -2.60000000000000021e43 < z < -6.40000000000000018e-305

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t around inf 77.1%

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

   if -6.40000000000000018e-305 < z < 7.2000000000000003e99

   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 inf 76.5%

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

      1. *-commutative 76.5%

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

   7. Simplified 76.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 6.8 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.4e+81) (not (<= z 6.8e+99))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e+81) || !(z <= 6.8e+99)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-3.4d+81)) .or. (.not. (z <= 6.8d+99))) then
        tmp = y / a
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e+81) || !(z <= 6.8e+99)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.4e+81) or not (z <= 6.8e+99):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.4e+81) || !(z <= 6.8e+99))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.4e+81) || ~((z <= 6.8e+99)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.4e+81], N[Not[LessEqual[z, 6.8e+99]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000003e81 or 6.79999999999999968e99 < z

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in z around inf 71.5%

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

   if -3.40000000000000003e81 < z < 6.79999999999999968e99

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in x around inf 69.2%

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

      1. *-commutative 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+81} \lor \neg \left(z \leq 6.8 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+37} \lor \neg \left(z \leq 1.35 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7e+37) (not (<= z 1.35e+82))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7e+37) || !(z <= 1.35e+82)) {
		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 <= (-7d+37)) .or. (.not. (z <= 1.35d+82))) 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 <= -7e+37) || !(z <= 1.35e+82)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7e+37) or not (z <= 1.35e+82):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7e+37) || !(z <= 1.35e+82))
		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 <= -7e+37) || ~((z <= 1.35e+82)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7e+37], N[Not[LessEqual[z, 1.35e+82]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7e37 or 1.35e82 < z

   1. Initial program 60.7%

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

      1. *-commutative 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in z around inf 65.3%

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

   if -7e37 < z < 1.35e82

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in z around 0 50.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+37} \lor \neg \left(z \leq 1.35 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% 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 84.4%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
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 z around 0 35.6%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
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 2024090 
(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))))