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

Percentage Accurate: 85.2% → 91.5%

Time: 14.3s

Alternatives: 8

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% 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: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+169}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e+169)
   (/ (- y) (- (/ t z) a))
   (if (<= z 1.5e+108) (/ (- x (* z y)) (- t (* z a))) (/ (- y (/ x z)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+169) {
		tmp = -y / ((t / z) - a);
	} else if (z <= 1.5e+108) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4d+169)) then
        tmp = -y / ((t / z) - a)
    else if (z <= 1.5d+108) then
        tmp = (x - (z * y)) / (t - (z * a))
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e+169) {
		tmp = -y / ((t / z) - a);
	} else if (z <= 1.5e+108) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4e+169:
		tmp = -y / ((t / z) - a)
	elif z <= 1.5e+108:
		tmp = (x - (z * y)) / (t - (z * a))
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e+169)
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	elseif (z <= 1.5e+108)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4e+169)
		tmp = -y / ((t / z) - a);
	elseif (z <= 1.5e+108)
		tmp = (x - (z * y)) / (t - (z * a));
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e+169], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+108], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999974e169

   1. Initial program 56.4%

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

      1. *-commutative 56.4%

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

   3. Simplified 56.4%

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

      1. clear-num 56.4%

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

      2. associate-/r/ 56.3%

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

   5. Applied egg-rr 56.3%

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

   6. Taylor expanded in x around 0 48.0%

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

      1. mul-1-neg 48.0%

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

      2. associate-/l* 64.9%

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

      3. distribute-neg-frac 64.9%

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

      4. *-commutative 64.9%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in t around 0 90.9%

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

       1. neg-mul-1 90.9%

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

       2. unsub-neg 90.9%

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

   11. Simplified 90.9%

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

   if -3.99999999999999974e169 < z < 1.49999999999999992e108

   1. Initial program 95.6%

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

   if 1.49999999999999992e108 < z

   1. Initial program 56.6%

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

      1. *-commutative 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in z around inf 75.3%

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

      1. +-commutative 75.3%

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

      2. associate--l+ 75.3%

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

      3. associate-/r* 75.2%

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

      4. associate-*r/ 75.2%

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

      5. associate-/r* 75.2%

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

      6. associate-*r/ 75.2%

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

      7. div-sub 75.2%

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

      8. distribute-lft-out-- 75.2%

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

      9. associate-*r/ 75.2%

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

      10. mul-1-neg 75.2%

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

      11. unsub-neg 75.2%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in a around inf 87.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+169}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 2: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -1.05e+96)
     (/ y a)
     (if (<= z -1.75e-48)
       t_1
       (if (<= z 1.9e+66)
         (/ (- x (* z y)) t)
         (if (<= z 1.7e+116) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.05e+96) {
		tmp = y / a;
	} else if (z <= -1.75e-48) {
		tmp = t_1;
	} else if (z <= 1.9e+66) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.7e+116) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    if (z <= (-1.05d+96)) then
        tmp = y / a
    else if (z <= (-1.75d-48)) then
        tmp = t_1
    else if (z <= 1.9d+66) then
        tmp = (x - (z * y)) / t
    else if (z <= 1.7d+116) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.05e+96) {
		tmp = y / a;
	} else if (z <= -1.75e-48) {
		tmp = t_1;
	} else if (z <= 1.9e+66) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.7e+116) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if z <= -1.05e+96:
		tmp = y / a
	elif z <= -1.75e-48:
		tmp = t_1
	elif z <= 1.9e+66:
		tmp = (x - (z * y)) / t
	elif z <= 1.7e+116:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (z <= -1.05e+96)
		tmp = Float64(y / a);
	elseif (z <= -1.75e-48)
		tmp = t_1;
	elseif (z <= 1.9e+66)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1.7e+116)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (z <= -1.05e+96)
		tmp = y / a;
	elseif (z <= -1.75e-48)
		tmp = t_1;
	elseif (z <= 1.9e+66)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1.7e+116)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e+96], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.75e-48], t$95$1, If[LessEqual[z, 1.9e+66], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.7e+116], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0500000000000001e96 or 1.70000000000000011e116 < z

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 70.9%

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

   if -1.0500000000000001e96 < z < -1.74999999999999996e-48 or 1.9000000000000001e66 < z < 1.70000000000000011e116

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around inf 68.4%

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

      1. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if -1.74999999999999996e-48 < z < 1.9000000000000001e66

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in t around inf 78.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-x}{a}}{z}\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6600000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) a) z)))
   (if (<= z -7.6e+87)
     (/ y a)
     (if (<= z -5e-25)
       t_1
       (if (<= z 6600000000000.0)
         (/ x t)
         (if (<= z 1.65e+116) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-x / a) / z;
	double tmp;
	if (z <= -7.6e+87) {
		tmp = y / a;
	} else if (z <= -5e-25) {
		tmp = t_1;
	} else if (z <= 6600000000000.0) {
		tmp = x / t;
	} else if (z <= 1.65e+116) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / a) / z
    if (z <= (-7.6d+87)) then
        tmp = y / a
    else if (z <= (-5d-25)) then
        tmp = t_1
    else if (z <= 6600000000000.0d0) then
        tmp = x / t
    else if (z <= 1.65d+116) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (-x / a) / z;
	double tmp;
	if (z <= -7.6e+87) {
		tmp = y / a;
	} else if (z <= -5e-25) {
		tmp = t_1;
	} else if (z <= 6600000000000.0) {
		tmp = x / t;
	} else if (z <= 1.65e+116) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (-x / a) / z
	tmp = 0
	if z <= -7.6e+87:
		tmp = y / a
	elif z <= -5e-25:
		tmp = t_1
	elif z <= 6600000000000.0:
		tmp = x / t
	elif z <= 1.65e+116:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-x) / a) / z)
	tmp = 0.0
	if (z <= -7.6e+87)
		tmp = Float64(y / a);
	elseif (z <= -5e-25)
		tmp = t_1;
	elseif (z <= 6600000000000.0)
		tmp = Float64(x / t);
	elseif (z <= 1.65e+116)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (-x / a) / z;
	tmp = 0.0;
	if (z <= -7.6e+87)
		tmp = y / a;
	elseif (z <= -5e-25)
		tmp = t_1;
	elseif (z <= 6600000000000.0)
		tmp = x / t;
	elseif (z <= 1.65e+116)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-x) / a), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -7.6e+87], N[(y / a), $MachinePrecision], If[LessEqual[z, -5e-25], t$95$1, If[LessEqual[z, 6600000000000.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.65e+116], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.60000000000000022e87 or 1.6499999999999999e116 < z

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 70.9%

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

   if -7.60000000000000022e87 < z < -4.99999999999999962e-25 or 6.6e12 < z < 1.6499999999999999e116

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in x around inf 62.7%

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

      1. *-commutative 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in t around 0 51.0%

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

      1. mul-1-neg 51.0%

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

      2. associate-/r* 56.0%

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

   9. Simplified 56.0%

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

   if -4.99999999999999962e-25 < z < 6.6e12

   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. Taylor expanded in z around 0 62.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 6600000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 13000000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.6e+87)
   (/ y a)
   (if (<= z -1.65e-24)
     (/ (/ (- x) a) z)
     (if (<= z 13000000000000.0)
       (/ x t)
       (if (<= z 1.65e+116) (/ (/ (- x) z) a) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.6e+87) {
		tmp = y / a;
	} else if (z <= -1.65e-24) {
		tmp = (-x / a) / z;
	} else if (z <= 13000000000000.0) {
		tmp = x / t;
	} else if (z <= 1.65e+116) {
		tmp = (-x / 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 <= (-7.6d+87)) then
        tmp = y / a
    else if (z <= (-1.65d-24)) then
        tmp = (-x / a) / z
    else if (z <= 13000000000000.0d0) then
        tmp = x / t
    else if (z <= 1.65d+116) then
        tmp = (-x / 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 <= -7.6e+87) {
		tmp = y / a;
	} else if (z <= -1.65e-24) {
		tmp = (-x / a) / z;
	} else if (z <= 13000000000000.0) {
		tmp = x / t;
	} else if (z <= 1.65e+116) {
		tmp = (-x / z) / a;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.6e+87:
		tmp = y / a
	elif z <= -1.65e-24:
		tmp = (-x / a) / z
	elif z <= 13000000000000.0:
		tmp = x / t
	elif z <= 1.65e+116:
		tmp = (-x / z) / a
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.6e+87)
		tmp = Float64(y / a);
	elseif (z <= -1.65e-24)
		tmp = Float64(Float64(Float64(-x) / a) / z);
	elseif (z <= 13000000000000.0)
		tmp = Float64(x / t);
	elseif (z <= 1.65e+116)
		tmp = Float64(Float64(Float64(-x) / 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 <= -7.6e+87)
		tmp = y / a;
	elseif (z <= -1.65e-24)
		tmp = (-x / a) / z;
	elseif (z <= 13000000000000.0)
		tmp = x / t;
	elseif (z <= 1.65e+116)
		tmp = (-x / z) / a;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.6e+87], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.65e-24], N[(N[((-x) / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 13000000000000.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.65e+116], N[(N[((-x) / z), $MachinePrecision] / a), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.60000000000000022e87 or 1.6499999999999999e116 < z

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 70.9%

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

   if -7.60000000000000022e87 < z < -1.64999999999999992e-24

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around inf 57.7%

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

      1. *-commutative 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in t around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. associate-/r* 53.7%

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

   9. Simplified 53.7%

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

   if -1.64999999999999992e-24 < z < 1.3e13

   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. Taylor expanded in z around 0 62.7%

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

   if 1.3e13 < z < 1.6499999999999999e116

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in x around inf 67.9%

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

      1. *-commutative 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in t around 0 53.0%

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

      1. associate-*r/ 53.0%

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

      2. neg-mul-1 53.0%

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

      3. *-commutative 53.0%

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

      4. associate-/r* 62.9%

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

   9. Simplified 62.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq 13000000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -32 \lor \neg \left(z \leq 8500000000000\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -32.0) (not (<= z 8500000000000.0)))
   (/ (- y (/ x z)) a)
   (/ (- x (* z y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -32.0) || !(z <= 8500000000000.0)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-32.0d0)) .or. (.not. (z <= 8500000000000.0d0))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (z * y)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -32.0) || !(z <= 8500000000000.0)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -32.0) or not (z <= 8500000000000.0):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (z * y)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -32.0) || !(z <= 8500000000000.0))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -32.0) || ~((z <= 8500000000000.0)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (z * y)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -32.0], N[Not[LessEqual[z, 8500000000000.0]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -32 or 8.5e12 < z

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in z around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. associate--l+ 69.7%

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

      3. associate-/r* 69.1%

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

      4. associate-*r/ 69.1%

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

      5. associate-/r* 69.1%

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

      6. associate-*r/ 69.1%

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

      7. div-sub 69.1%

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

      8. distribute-lft-out-- 69.1%

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

      9. associate-*r/ 69.1%

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

      10. mul-1-neg 69.1%

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

      11. unsub-neg 69.1%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in a around inf 82.0%

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

   if -32 < z < 8.5e12

   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. Taylor expanded in t around inf 78.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -32 \lor \neg \left(z \leq 8500000000000\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \]

Alternative 6: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+79) (/ y a) (if (<= z 1.8e+116) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+79) {
		tmp = y / a;
	} else if (z <= 1.8e+116) {
		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 <= (-5d+79)) then
        tmp = y / a
    else if (z <= 1.8d+116) 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 <= -5e+79) {
		tmp = y / a;
	} else if (z <= 1.8e+116) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e+79:
		tmp = y / a
	elif z <= 1.8e+116:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+79)
		tmp = Float64(y / a);
	elseif (z <= 1.8e+116)
		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 <= -5e+79)
		tmp = y / a;
	elseif (z <= 1.8e+116)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+79], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e+116], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e79 or 1.79999999999999985e116 < z

   1. Initial program 59.0%

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

      1. *-commutative 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in z around inf 70.9%

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

   if -5e79 < z < 1.79999999999999985e116

   1. Initial program 96.4%

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

      1. *-commutative 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in x around inf 72.4%

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

      1. *-commutative 72.4%

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

   6. Simplified 72.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7: 54.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.7e-26) (/ y a) (if (<= z 1.6e+98) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.7e-26) {
		tmp = y / a;
	} else if (z <= 1.6e+98) {
		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 <= (-3.7d-26)) then
        tmp = y / a
    else if (z <= 1.6d+98) 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 <= -3.7e-26) {
		tmp = y / a;
	} else if (z <= 1.6e+98) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.7e-26:
		tmp = y / a
	elif z <= 1.6e+98:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.7e-26)
		tmp = Float64(y / a);
	elseif (z <= 1.6e+98)
		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 <= -3.7e-26)
		tmp = y / a;
	elseif (z <= 1.6e+98)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.7e-26], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.6e+98], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6999999999999999e-26 or 1.6000000000000001e98 < z

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in z around inf 59.2%

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

   if -3.6999999999999999e-26 < z < 1.6000000000000001e98

   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. Taylor expanded in z around 0 60.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8: 35.1% 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 84.5%

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

   1. *-commutative 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in z around 0 38.6%

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

5. Final simplification 38.6%

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

Developer target: 97.1% accurate, 0.6× 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 2023274 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (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))))