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

Percentage Accurate: 96.1% → 99.7%

Time: 8.4s

Alternatives: 6

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 5e+241)))
   (/ (/ (- x) z) t)
   (/ x (- y (* z t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 5e+241)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 5e+241)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 5e+241):
		tmp = (-x / z) / t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 5e+241))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 5e+241)))
		tmp = (-x / z) / t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+241]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 5.00000000000000025e241 < (*.f64 z t)

   1. Initial program 64.9%

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

      1. clear-num 64.9%

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

      2. associate-/r/ 64.9%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in t around inf 78.5%

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

      1. distribute-lft-out 78.5%

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

      2. associate-*r/ 78.5%

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

      3. mul-1-neg 78.5%

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

      4. associate-/r* 78.5%

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

   7. Simplified 78.5%

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

   8. Taylor expanded in z around inf 99.8%

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

   if -inf.0 < (*.f64 z t) < 5.00000000000000025e241

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+261}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) z) t)))
   (if (<= (* z t) -5e+23)
     t_1
     (if (<= (* z t) 2e-124)
       (/ x y)
       (if (<= (* z t) 1e+261) (* x (/ -1.0 (* z t))) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e-124) {
		tmp = x / y;
	} else if ((z * t) <= 1e+261) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / z) / t
    if ((z * t) <= (-5d+23)) then
        tmp = t_1
    else if ((z * t) <= 2d-124) then
        tmp = x / y
    else if ((z * t) <= 1d+261) then
        tmp = x * ((-1.0d0) / (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e-124) {
		tmp = x / y;
	} else if ((z * t) <= 1e+261) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (-x / z) / t
	tmp = 0
	if (z * t) <= -5e+23:
		tmp = t_1
	elif (z * t) <= 2e-124:
		tmp = x / y
	elif (z * t) <= 1e+261:
		tmp = x * (-1.0 / (z * t))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (Float64(z * t) <= -5e+23)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-124)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e+261)
		tmp = Float64(x * Float64(-1.0 / Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (-x / z) / t;
	tmp = 0.0;
	if ((z * t) <= -5e+23)
		tmp = t_1;
	elseif ((z * t) <= 2e-124)
		tmp = x / y;
	elseif ((z * t) <= 1e+261)
		tmp = x * (-1.0 / (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+23], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-124], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+261], N[(x * N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 9.9999999999999993e260 < (*.f64 z t)

   1. Initial program 81.3%

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

      1. clear-num 80.6%

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

      2. associate-/r/ 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in t around inf 73.1%

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

      1. distribute-lft-out 73.1%

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

      2. associate-*r/ 73.1%

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

      3. mul-1-neg 73.1%

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

      4. associate-/r* 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in z around inf 88.3%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.99999999999999987e-124

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.5%

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

   if 1.99999999999999987e-124 < (*.f64 z t) < 9.9999999999999993e260

   1. Initial program 99.7%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 75.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+261}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) z) t)))
   (if (<= (* z t) -5e+23)
     t_1
     (if (<= (* z t) 2e-124)
       (/ x y)
       (if (<= (* z t) 5e+241) (/ x (- (* z t))) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e-124) {
		tmp = x / y;
	} else if ((z * t) <= 5e+241) {
		tmp = x / -(z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / z) / t
    if ((z * t) <= (-5d+23)) then
        tmp = t_1
    else if ((z * t) <= 2d-124) then
        tmp = x / y
    else if ((z * t) <= 5d+241) then
        tmp = x / -(z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e-124) {
		tmp = x / y;
	} else if ((z * t) <= 5e+241) {
		tmp = x / -(z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (-x / z) / t
	tmp = 0
	if (z * t) <= -5e+23:
		tmp = t_1
	elif (z * t) <= 2e-124:
		tmp = x / y
	elif (z * t) <= 5e+241:
		tmp = x / -(z * t)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (Float64(z * t) <= -5e+23)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-124)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e+241)
		tmp = Float64(x / Float64(-Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (-x / z) / t;
	tmp = 0.0;
	if ((z * t) <= -5e+23)
		tmp = t_1;
	elseif ((z * t) <= 2e-124)
		tmp = x / y;
	elseif ((z * t) <= 5e+241)
		tmp = x / -(z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+23], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-124], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+241], N[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 5.00000000000000025e241 < (*.f64 z t)

   1. Initial program 82.0%

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

      1. clear-num 81.3%

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

      2. associate-/r/ 81.9%

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

   4. Applied egg-rr 81.9%

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

   5. Taylor expanded in t around inf 74.0%

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

      1. distribute-lft-out 74.0%

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

      2. associate-*r/ 74.0%

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

      3. mul-1-neg 74.0%

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

      4. associate-/r* 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in z around inf 88.7%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.99999999999999987e-124

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.5%

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

   if 1.99999999999999987e-124 < (*.f64 z t) < 5.00000000000000025e241

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. neg-mul-1 74.1%

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

   5. Simplified 74.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+23) (not (<= (* z t) 2e-124)))
   (/ (/ (- x) z) t)
   (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 2e-124)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-5d+23)) .or. (.not. ((z * t) <= 2d-124))) then
        tmp = (-x / z) / t
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 2e-124)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+23) or not ((z * t) <= 2e-124):
		tmp = (-x / z) / t
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+23) || !(Float64(z * t) <= 2e-124))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+23) || ~(((z * t) <= 2e-124)))
		tmp = (-x / z) / t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+23], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-124]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 1.99999999999999987e-124 < (*.f64 z t)

   1. Initial program 89.3%

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

      1. clear-num 88.4%

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

      2. associate-/r/ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in t around inf 62.4%

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

      1. distribute-lft-out 62.4%

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

      2. associate-*r/ 62.4%

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

      3. mul-1-neg 62.4%

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

      4. associate-/r* 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in z around inf 76.8%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.99999999999999987e-124

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+165} \lor \neg \left(z \cdot t \leq 10^{+216}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -2e+165) (not (<= (* z t) 1e+216)))
   (/ x (* z t))
   (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+165) || !((z * t) <= 1e+216)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-2d+165)) .or. (.not. ((z * t) <= 1d+216))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+165) || !((z * t) <= 1e+216)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -2e+165) or not ((z * t) <= 1e+216):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+165) || !(Float64(z * t) <= 1e+216))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -2e+165) || ~(((z * t) <= 1e+216)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+165], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+216]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.9999999999999998e165 or 1e216 < (*.f64 z t)

   1. Initial program 76.6%

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

   3. Taylor expanded in z around -inf 78.0%

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

   4. Taylor expanded in t around inf 72.0%

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

      1. associate-/r* 95.2%

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

   6. Simplified 95.2%

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

      1. add-sqr-sqrt 71.4%

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

      2. sqrt-unprod 59.3%

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

      3. mul-1-neg 59.3%

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

      4. mul-1-neg 59.3%

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

      5. sqr-neg 59.3%

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

      6. sqrt-unprod 49.3%

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

      7. add-sqr-sqrt 51.1%

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

      8. associate-/l/ 51.6%

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

   8. Applied egg-rr 51.6%

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

   if -1.9999999999999998e165 < (*.f64 z t) < 1e216

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 65.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+165} \lor \neg \left(z \cdot t \leq 10^{+216}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return x / y;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / y
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x / y

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

3. Taylor expanded in y around inf 54.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Add Preprocessing

Developer Target 1: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024145 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -161819597360704900000000000000000000000000000000000) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 213783064348764440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t))))))

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