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

Percentage Accurate: 84.8% → 90.4%

Time: 13.7s

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: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+98} \lor \neg \left(z \leq 2.3 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.4e+98) (not (<= z 2.3e+72)))
   (/ (- y (/ x z)) a)
   (/ (- x (* z y)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.4e+98) || !(z <= 2.3e+72)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (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 <= (-4.4d+98)) .or. (.not. (z <= 2.3d+72))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (z * y)) / (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 <= -4.4e+98) || !(z <= 2.3e+72)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.4e+98) or not (z <= 2.3e+72):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (z * y)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.4e+98) || !(z <= 2.3e+72))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.4e+98) || ~((z <= 2.3e+72)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (z * y)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.4e+98], N[Not[LessEqual[z, 2.3e+72]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.40000000000000017e98 or 2.3e72 < z

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around inf 50.0%

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

      1. +-commutative 50.0%

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

      2. mul-1-neg 50.0%

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

      3. unsub-neg 50.0%

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

      4. *-commutative 50.0%

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

      5. associate-/l* 51.6%

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

      6. *-commutative 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in a around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. associate-*r* 74.3%

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

      3. mul-1-neg 74.3%

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

      4. +-commutative 74.3%

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

      5. mul-1-neg 74.3%

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

      6. sub-neg 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in x around 0 81.3%

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

       1. mul-1-neg 81.3%

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

       2. unsub-neg 81.3%

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

   13. Simplified 81.3%

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

   if -4.40000000000000017e98 < z < 2.3e72

   1. Initial program 99.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+98} \lor \neg \left(z \leq 2.3 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+116}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 82000000000000:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+116)
   (/ y a)
   (if (<= z -8e+57)
     (/ (/ x z) (- a))
     (if (<= z -1.36e-27)
       (/ y a)
       (if (<= z 8.8e-47)
         (/ x t)
         (if (<= z 82000000000000.0)
           (* x (/ (/ -1.0 a) z))
           (if (<= z 6.8e+25) (/ x t) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+116) {
		tmp = y / a;
	} else if (z <= -8e+57) {
		tmp = (x / z) / -a;
	} else if (z <= -1.36e-27) {
		tmp = y / a;
	} else if (z <= 8.8e-47) {
		tmp = x / t;
	} else if (z <= 82000000000000.0) {
		tmp = x * ((-1.0 / a) / z);
	} else if (z <= 6.8e+25) {
		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 <= (-2.3d+116)) then
        tmp = y / a
    else if (z <= (-8d+57)) then
        tmp = (x / z) / -a
    else if (z <= (-1.36d-27)) then
        tmp = y / a
    else if (z <= 8.8d-47) then
        tmp = x / t
    else if (z <= 82000000000000.0d0) then
        tmp = x * (((-1.0d0) / a) / z)
    else if (z <= 6.8d+25) 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 <= -2.3e+116) {
		tmp = y / a;
	} else if (z <= -8e+57) {
		tmp = (x / z) / -a;
	} else if (z <= -1.36e-27) {
		tmp = y / a;
	} else if (z <= 8.8e-47) {
		tmp = x / t;
	} else if (z <= 82000000000000.0) {
		tmp = x * ((-1.0 / a) / z);
	} else if (z <= 6.8e+25) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+116:
		tmp = y / a
	elif z <= -8e+57:
		tmp = (x / z) / -a
	elif z <= -1.36e-27:
		tmp = y / a
	elif z <= 8.8e-47:
		tmp = x / t
	elif z <= 82000000000000.0:
		tmp = x * ((-1.0 / a) / z)
	elif z <= 6.8e+25:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+116)
		tmp = Float64(y / a);
	elseif (z <= -8e+57)
		tmp = Float64(Float64(x / z) / Float64(-a));
	elseif (z <= -1.36e-27)
		tmp = Float64(y / a);
	elseif (z <= 8.8e-47)
		tmp = Float64(x / t);
	elseif (z <= 82000000000000.0)
		tmp = Float64(x * Float64(Float64(-1.0 / a) / z));
	elseif (z <= 6.8e+25)
		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 <= -2.3e+116)
		tmp = y / a;
	elseif (z <= -8e+57)
		tmp = (x / z) / -a;
	elseif (z <= -1.36e-27)
		tmp = y / a;
	elseif (z <= 8.8e-47)
		tmp = x / t;
	elseif (z <= 82000000000000.0)
		tmp = x * ((-1.0 / a) / z);
	elseif (z <= 6.8e+25)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+116], N[(y / a), $MachinePrecision], If[LessEqual[z, -8e+57], N[(N[(x / z), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[z, -1.36e-27], N[(y / a), $MachinePrecision], If[LessEqual[z, 8.8e-47], N[(x / t), $MachinePrecision], If[LessEqual[z, 82000000000000.0], N[(x * N[(N[(-1.0 / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+25], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.29999999999999995e116 or -8.00000000000000039e57 < z < -1.36e-27 or 6.79999999999999967e25 < z

   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 z around inf 62.8%

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

   if -2.29999999999999995e116 < z < -8.00000000000000039e57

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in x around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. mul-1-neg 70.0%

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

      3. unsub-neg 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.7%

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

      6. *-commutative 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in a around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. mul-1-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. mul-1-neg 69.9%

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

      6. sub-neg 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in x around inf 63.2%

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

       1. associate-*r/ 63.2%

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

       2. mul-1-neg 63.2%

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

   13. Simplified 63.2%

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

   if -1.36e-27 < z < 8.80000000000000075e-47 or 8.2e13 < z < 6.79999999999999967e25

   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 63.7%

      \[\leadsto \color{blue}{\frac{x}{t}} \]

   if 8.80000000000000075e-47 < z < 8.2e13

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 92.6%

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

      1. *-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. div-inv 92.5%

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

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in t around 0 64.8%

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

       1. associate-/r* 64.8%

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

   12. Simplified 64.8%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{-1}{a}}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+116}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 82000000000000:\\ \;\;\;\;x \cdot \frac{\frac{-1}{a}}{z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 60000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+115)
   (/ y a)
   (if (<= z -1.05e+58)
     (/ (/ x z) (- a))
     (if (<= z -2.9e-27)
       (/ y a)
       (if (<= z 6.2e-47)
         (/ x t)
         (if (<= z 60000000000.0)
           (/ (- x) (* z a))
           (if (<= z 4.7e+26) (/ x t) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+115) {
		tmp = y / a;
	} else if (z <= -1.05e+58) {
		tmp = (x / z) / -a;
	} else if (z <= -2.9e-27) {
		tmp = y / a;
	} else if (z <= 6.2e-47) {
		tmp = x / t;
	} else if (z <= 60000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 4.7e+26) {
		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 <= (-2.8d+115)) then
        tmp = y / a
    else if (z <= (-1.05d+58)) then
        tmp = (x / z) / -a
    else if (z <= (-2.9d-27)) then
        tmp = y / a
    else if (z <= 6.2d-47) then
        tmp = x / t
    else if (z <= 60000000000.0d0) then
        tmp = -x / (z * a)
    else if (z <= 4.7d+26) 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 <= -2.8e+115) {
		tmp = y / a;
	} else if (z <= -1.05e+58) {
		tmp = (x / z) / -a;
	} else if (z <= -2.9e-27) {
		tmp = y / a;
	} else if (z <= 6.2e-47) {
		tmp = x / t;
	} else if (z <= 60000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 4.7e+26) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.8e+115:
		tmp = y / a
	elif z <= -1.05e+58:
		tmp = (x / z) / -a
	elif z <= -2.9e-27:
		tmp = y / a
	elif z <= 6.2e-47:
		tmp = x / t
	elif z <= 60000000000.0:
		tmp = -x / (z * a)
	elif z <= 4.7e+26:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+115)
		tmp = Float64(y / a);
	elseif (z <= -1.05e+58)
		tmp = Float64(Float64(x / z) / Float64(-a));
	elseif (z <= -2.9e-27)
		tmp = Float64(y / a);
	elseif (z <= 6.2e-47)
		tmp = Float64(x / t);
	elseif (z <= 60000000000.0)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 4.7e+26)
		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 <= -2.8e+115)
		tmp = y / a;
	elseif (z <= -1.05e+58)
		tmp = (x / z) / -a;
	elseif (z <= -2.9e-27)
		tmp = y / a;
	elseif (z <= 6.2e-47)
		tmp = x / t;
	elseif (z <= 60000000000.0)
		tmp = -x / (z * a);
	elseif (z <= 4.7e+26)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+115], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.05e+58], N[(N[(x / z), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[z, -2.9e-27], N[(y / a), $MachinePrecision], If[LessEqual[z, 6.2e-47], N[(x / t), $MachinePrecision], If[LessEqual[z, 60000000000.0], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.7e+26], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.8e115 or -1.05000000000000006e58 < z < -2.90000000000000004e-27 or 4.6999999999999998e26 < z

   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 z around inf 62.8%

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

   if -2.8e115 < z < -1.05000000000000006e58

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in x around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. mul-1-neg 70.0%

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

      3. unsub-neg 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.7%

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

      6. *-commutative 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in a around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. mul-1-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. mul-1-neg 69.9%

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

      6. sub-neg 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in x around inf 63.2%

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

       1. associate-*r/ 63.2%

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

       2. mul-1-neg 63.2%

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

   13. Simplified 63.2%

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

   if -2.90000000000000004e-27 < z < 6.1999999999999996e-47 or 6e10 < z < 4.6999999999999998e26

   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 63.7%

      \[\leadsto \color{blue}{\frac{x}{t}} \]

   if 6.1999999999999996e-47 < z < 6e10

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. neg-mul-1 71.5%

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

      3. neg-sub0 71.5%

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

      4. sub-neg 71.5%

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

      5. distribute-rgt-neg-out 71.5%

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

      6. +-commutative 71.5%

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

      7. associate--r+ 71.5%

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

      8. neg-sub0 71.5%

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

      9. distribute-rgt-neg-out 71.5%

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

      10. remove-double-neg 71.5%

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

      11. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in y around 0 64.7%

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

      1. mul-1-neg 64.7%

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

   10. Simplified 64.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{x}{z}}{-a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 60000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{a}}{-z}\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 5500000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e+115)
   (/ y a)
   (if (<= z -3.5e+57)
     (/ (/ x a) (- z))
     (if (<= z -1.36e-27)
       (/ y a)
       (if (<= z 1.4e-46)
         (/ x t)
         (if (<= z 5500000000000.0)
           (/ (- x) (* z a))
           (if (<= z 1.26e+26) (/ x t) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e+115) {
		tmp = y / a;
	} else if (z <= -3.5e+57) {
		tmp = (x / a) / -z;
	} else if (z <= -1.36e-27) {
		tmp = y / a;
	} else if (z <= 1.4e-46) {
		tmp = x / t;
	} else if (z <= 5500000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 1.26e+26) {
		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 <= (-4.6d+115)) then
        tmp = y / a
    else if (z <= (-3.5d+57)) then
        tmp = (x / a) / -z
    else if (z <= (-1.36d-27)) then
        tmp = y / a
    else if (z <= 1.4d-46) then
        tmp = x / t
    else if (z <= 5500000000000.0d0) then
        tmp = -x / (z * a)
    else if (z <= 1.26d+26) 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 <= -4.6e+115) {
		tmp = y / a;
	} else if (z <= -3.5e+57) {
		tmp = (x / a) / -z;
	} else if (z <= -1.36e-27) {
		tmp = y / a;
	} else if (z <= 1.4e-46) {
		tmp = x / t;
	} else if (z <= 5500000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 1.26e+26) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.6e+115:
		tmp = y / a
	elif z <= -3.5e+57:
		tmp = (x / a) / -z
	elif z <= -1.36e-27:
		tmp = y / a
	elif z <= 1.4e-46:
		tmp = x / t
	elif z <= 5500000000000.0:
		tmp = -x / (z * a)
	elif z <= 1.26e+26:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e+115)
		tmp = Float64(y / a);
	elseif (z <= -3.5e+57)
		tmp = Float64(Float64(x / a) / Float64(-z));
	elseif (z <= -1.36e-27)
		tmp = Float64(y / a);
	elseif (z <= 1.4e-46)
		tmp = Float64(x / t);
	elseif (z <= 5500000000000.0)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 1.26e+26)
		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 <= -4.6e+115)
		tmp = y / a;
	elseif (z <= -3.5e+57)
		tmp = (x / a) / -z;
	elseif (z <= -1.36e-27)
		tmp = y / a;
	elseif (z <= 1.4e-46)
		tmp = x / t;
	elseif (z <= 5500000000000.0)
		tmp = -x / (z * a);
	elseif (z <= 1.26e+26)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e+115], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.5e+57], N[(N[(x / a), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[z, -1.36e-27], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.4e-46], N[(x / t), $MachinePrecision], If[LessEqual[z, 5500000000000.0], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.26e+26], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.60000000000000007e115 or -3.4999999999999997e57 < z < -1.36e-27 or 1.25999999999999995e26 < z

   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 z around inf 62.8%

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

   if -4.60000000000000007e115 < z < -3.4999999999999997e57

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in x around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. mul-1-neg 70.0%

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

      3. unsub-neg 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.7%

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

      6. *-commutative 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in a around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. mul-1-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. mul-1-neg 69.9%

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

      6. sub-neg 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in x around 0 69.9%

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

       1. mul-1-neg 69.9%

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

       2. unsub-neg 69.9%

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

   13. Simplified 69.9%

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

   14. Taylor expanded in y around 0 48.4%

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

       1. mul-1-neg 48.4%

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

       2. associate-/r* 63.1%

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

       3. distribute-frac-neg2 63.1%

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

   16. Simplified 63.1%

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

   if -1.36e-27 < z < 1.3999999999999999e-46 or 5.5e12 < z < 1.25999999999999995e26

   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 63.7%

      \[\leadsto \color{blue}{\frac{x}{t}} \]

   if 1.3999999999999999e-46 < z < 5.5e12

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. neg-mul-1 71.5%

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

      3. neg-sub0 71.5%

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

      4. sub-neg 71.5%

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

      5. distribute-rgt-neg-out 71.5%

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

      6. +-commutative 71.5%

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

      7. associate--r+ 71.5%

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

      8. neg-sub0 71.5%

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

      9. distribute-rgt-neg-out 71.5%

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

      10. remove-double-neg 71.5%

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

      11. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in y around 0 64.7%

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

      1. mul-1-neg 64.7%

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

   10. Simplified 64.7%

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

Alternative 5: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 35000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-27)
   (/ y a)
   (if (<= z 7.2e-47)
     (/ x t)
     (if (<= z 35000000000.0)
       (/ (- x) (* z a))
       (if (<= z 4.4e+26) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-27) {
		tmp = y / a;
	} else if (z <= 7.2e-47) {
		tmp = x / t;
	} else if (z <= 35000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 4.4e+26) {
		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 <= (-1d-27)) then
        tmp = y / a
    else if (z <= 7.2d-47) then
        tmp = x / t
    else if (z <= 35000000000.0d0) then
        tmp = -x / (z * a)
    else if (z <= 4.4d+26) 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 <= -1e-27) {
		tmp = y / a;
	} else if (z <= 7.2e-47) {
		tmp = x / t;
	} else if (z <= 35000000000.0) {
		tmp = -x / (z * a);
	} else if (z <= 4.4e+26) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-27:
		tmp = y / a
	elif z <= 7.2e-47:
		tmp = x / t
	elif z <= 35000000000.0:
		tmp = -x / (z * a)
	elif z <= 4.4e+26:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-27)
		tmp = Float64(y / a);
	elseif (z <= 7.2e-47)
		tmp = Float64(x / t);
	elseif (z <= 35000000000.0)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 4.4e+26)
		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 <= -1e-27)
		tmp = y / a;
	elseif (z <= 7.2e-47)
		tmp = x / t;
	elseif (z <= 35000000000.0)
		tmp = -x / (z * a);
	elseif (z <= 4.4e+26)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-27], N[(y / a), $MachinePrecision], If[LessEqual[z, 7.2e-47], N[(x / t), $MachinePrecision], If[LessEqual[z, 35000000000.0], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e+26], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e-27 or 4.40000000000000014e26 < z

   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 z around inf 59.2%

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

   if -1e-27 < z < 7.19999999999999982e-47 or 3.5e10 < z < 4.40000000000000014e26

   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 63.7%

      \[\leadsto \color{blue}{\frac{x}{t}} \]

   if 7.19999999999999982e-47 < z < 3.5e10

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. neg-mul-1 71.5%

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

      3. neg-sub0 71.5%

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

      4. sub-neg 71.5%

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

      5. distribute-rgt-neg-out 71.5%

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

      6. +-commutative 71.5%

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

      7. associate--r+ 71.5%

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

      8. neg-sub0 71.5%

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

      9. distribute-rgt-neg-out 71.5%

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

      10. remove-double-neg 71.5%

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

      11. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in y around 0 64.7%

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

      1. mul-1-neg 64.7%

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

   10. Simplified 64.7%

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

Alternative 6: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-43} \lor \neg \left(z \leq 1.2 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.4e-43) (not (<= z 1.2e+15)))
   (/ (- y (/ x z)) a)
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.4e-43) || !(z <= 1.2e+15)) {
		tmp = (y - (x / z)) / 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 <= (-2.4d-43)) .or. (.not. (z <= 1.2d+15))) then
        tmp = (y - (x / z)) / 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 <= -2.4e-43) || !(z <= 1.2e+15)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.4e-43) or not (z <= 1.2e+15):
		tmp = (y - (x / z)) / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.4e-43) || !(z <= 1.2e+15))
		tmp = Float64(Float64(y - Float64(x / z)) / 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 <= -2.4e-43) || ~((z <= 1.2e+15)))
		tmp = (y - (x / z)) / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.4e-43], N[Not[LessEqual[z, 1.2e+15]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000002e-43 or 1.2e15 < z

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. +-commutative 59.6%

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

      2. mul-1-neg 59.6%

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

      3. unsub-neg 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-/l* 60.1%

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

      6. *-commutative 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in a around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. associate-*r* 73.4%

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

      3. mul-1-neg 73.4%

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

      4. +-commutative 73.4%

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

      5. mul-1-neg 73.4%

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

      6. sub-neg 73.4%

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

   10. Simplified 73.4%

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

   11. Taylor expanded in x around 0 80.4%

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

       1. mul-1-neg 80.4%

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

       2. unsub-neg 80.4%

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

   13. Simplified 80.4%

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

   if -2.4000000000000002e-43 < z < 1.2e15

   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 81.9%

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

      1. *-commutative 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-43} \lor \neg \left(z \leq 1.2 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+117} \lor \neg \left(z \leq 1.7 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.3e+117) (not (<= z 1.7e+27))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.3e+117) || !(z <= 1.7e+27)) {
		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 <= (-4.3d+117)) .or. (.not. (z <= 1.7d+27))) 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 <= -4.3e+117) || !(z <= 1.7e+27)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.3e+117) or not (z <= 1.7e+27):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.3e+117) || !(z <= 1.7e+27))
		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 <= -4.3e+117) || ~((z <= 1.7e+27)))
		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, -4.3e+117], N[Not[LessEqual[z, 1.7e+27]], $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 < -4.29999999999999998e117 or 1.7e27 < z

   1. Initial program 61.2%

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

      1. *-commutative 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in z around inf 64.2%

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

   if -4.29999999999999998e117 < z < 1.7e27

   1. Initial program 97.3%

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

      1. *-commutative 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+117} \lor \neg \left(z \leq 1.7 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-27} \lor \neg \left(z \leq 4.2 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.55e-27) (not (<= z 4.2e+25))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.55e-27) || !(z <= 4.2e+25)) {
		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 <= (-1.55d-27)) .or. (.not. (z <= 4.2d+25))) 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 <= -1.55e-27) || !(z <= 4.2e+25)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.55e-27) or not (z <= 4.2e+25):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.55e-27) || !(z <= 4.2e+25))
		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 <= -1.55e-27) || ~((z <= 4.2e+25)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.55e-27], N[Not[LessEqual[z, 4.2e+25]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5499999999999999e-27 or 4.1999999999999998e25 < z

   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 z around inf 59.2%

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

   if -1.5499999999999999e-27 < z < 4.1999999999999998e25

   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 59.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-27} \lor \neg \left(z \leq 4.2 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.8% 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 82.9%

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

   1. *-commutative 82.9%

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

3. Simplified 82.9%

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

5. Taylor expanded in z around 0 35.0%

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

Developer target: 97.4% 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 2024100 
(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))))