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

Percentage Accurate: 95.8% → 99.8%

Time: 6.6s

Alternatives: 7

Speedup: 0.5×

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: 95.8% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+277}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{-1}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) z) t)
   (if (<= (* z t) 1e+277)
     (/ x (- y (* z t)))
     (/ 1.0 (/ z (* x (/ -1.0 t)))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+277) {
		tmp = x / (y - (z * t));
	} else {
		tmp = 1.0 / (z / (x * (-1.0 / t)));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+277) {
		tmp = x / (y - (z * t));
	} else {
		tmp = 1.0 / (z / (x * (-1.0 / t)));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-x / z) / t
	elif (z * t) <= 1e+277:
		tmp = x / (y - (z * t))
	else:
		tmp = 1.0 / (z / (x * (-1.0 / t)))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 1e+277)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(1.0 / Float64(z / Float64(x * Float64(-1.0 / t))));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 1e+277)
		tmp = x / (y - (z * t));
	else
		tmp = 1.0 / (z / (x * (-1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+277], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z / N[(x * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.7%

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

      1. clear-num 71.7%

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

      2. inv-pow 71.7%

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

   3. Applied egg-rr 71.7%

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

   4. Taylor expanded in y around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. associate-/l* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   6. Simplified 99.8%

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

      1. unpow-1 99.8%

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

      2. clear-num 99.8%

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

      3. frac-2neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. remove-double-neg 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -inf.0 < (*.f64 z t) < 1e277

   1. Initial program 99.9%

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

   if 1e277 < (*.f64 z t)

   1. Initial program 63.8%

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

      1. clear-num 63.8%

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

      2. associate-/r/ 63.8%

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

   3. Applied egg-rr 63.8%

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

   4. Taylor expanded in y around 0 63.8%

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

      1. associate-/r* 68.0%

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

   6. Simplified 68.0%

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

      1. associate-*l/ 99.8%

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

      2. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+277}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{-1}{t}}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+257}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: 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) 1e+257)))
   (/ (/ (- x) z) t)
   (/ x (- y (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 1e+257)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 1e+257)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 1e+257):
		tmp = (-x / z) / t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 1e+257))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 1e+257)))
		tmp = (-x / z) / t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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], 1e+257]], $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 1.00000000000000003e257 < (*.f64 z t)

   1. Initial program 69.2%

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

      1. clear-num 69.2%

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

      2. inv-pow 69.2%

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

   3. Applied egg-rr 69.2%

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

   4. Taylor expanded in y around 0 69.2%

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

      1. mul-1-neg 69.2%

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

      2. associate-/l* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   6. Simplified 99.8%

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

      1. unpow-1 99.8%

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. remove-double-neg 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000003e257

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+257}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 3: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-214} \lor \neg \left(t \leq 6 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.2e-214) (not (<= t 6e+78))) (/ (/ (- x) z) t) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.2e-214) || !(t <= 6e+78)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: 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 ((t <= (-7.2d-214)) .or. (.not. (t <= 6d+78))) then
        tmp = (-x / z) / t
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.2e-214) || !(t <= 6e+78)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -7.2e-214) or not (t <= 6e+78):
		tmp = (-x / z) / t
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -7.2e-214) || !(t <= 6e+78))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -7.2e-214) || ~((t <= 6e+78)))
		tmp = (-x / z) / t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -7.2e-214], N[Not[LessEqual[t, 6e+78]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2e-214 or 5.99999999999999964e78 < t

   1. Initial program 92.3%

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

      1. clear-num 91.5%

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

      2. inv-pow 91.5%

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

   3. Applied egg-rr 91.5%

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

   4. Taylor expanded in y around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. associate-/l* 67.9%

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

      3. distribute-neg-frac 67.9%

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

   6. Simplified 67.9%

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

      1. unpow-1 67.9%

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

      2. clear-num 68.5%

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

      3. frac-2neg 68.5%

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

      4. distribute-neg-frac 68.5%

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

      5. remove-double-neg 68.5%

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

   8. Applied egg-rr 68.5%

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

   if -7.2e-214 < t < 5.99999999999999964e78

   1. Initial program 98.9%

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

   2. Taylor expanded in y around inf 71.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-214} \lor \neg \left(t \leq 6 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.5e+17) (/ x y) (if (<= y 7.5e-61) (/ (- x) (* z t)) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e+17) {
		tmp = x / y;
	} else if (y <= 7.5e-61) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: 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 (y <= (-7.5d+17)) then
        tmp = x / y
    else if (y <= 7.5d-61) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e+17) {
		tmp = x / y;
	} else if (y <= 7.5e-61) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -7.5e+17:
		tmp = x / y
	elif y <= 7.5e-61:
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.5e+17)
		tmp = Float64(x / y);
	elseif (y <= 7.5e-61)
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -7.5e+17)
		tmp = x / y;
	elseif (y <= 7.5e-61)
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -7.5e+17], N[(x / y), $MachinePrecision], If[LessEqual[y, 7.5e-61], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e17 or 7.50000000000000047e-61 < y

   1. Initial program 95.6%

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

   2. Taylor expanded in y around inf 77.9%

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

   if -7.5e17 < y < 7.50000000000000047e-61

   1. Initial program 93.9%

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

   2. Taylor expanded in y around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. neg-mul-1 75.9%

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

   4. Simplified 75.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.8e-96)
   (/ (/ (- x) t) z)
   (if (<= t 2.15e+77) (/ x y) (/ (- x) (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.8e-96) {
		tmp = (-x / t) / z;
	} else if (t <= 2.15e+77) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Fortran

NOTE: 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 (t <= (-7.8d-96)) then
        tmp = (-x / t) / z
    else if (t <= 2.15d+77) then
        tmp = x / y
    else
        tmp = -x / (z * t)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.8e-96) {
		tmp = (-x / t) / z;
	} else if (t <= 2.15e+77) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -7.8e-96:
		tmp = (-x / t) / z
	elif t <= 2.15e+77:
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.8e-96)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (t <= 2.15e+77)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-x) / Float64(z * t));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.8e-96)
		tmp = (-x / t) / z;
	elseif (t <= 2.15e+77)
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -7.8e-96], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 2.15e+77], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.7999999999999997e-96

   1. Initial program 94.6%

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

      1. clear-num 93.3%

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

      2. associate-/r/ 94.4%

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

   3. Applied egg-rr 94.4%

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

   4. Taylor expanded in y around 0 61.0%

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

      1. associate-/r* 60.9%

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

   6. Simplified 60.9%

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

      1. associate-*l/ 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. associate-*l/ 66.9%

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

      2. neg-mul-1 66.9%

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

   10. Applied egg-rr 66.9%

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

   if -7.7999999999999997e-96 < t < 2.14999999999999996e77

   1. Initial program 99.1%

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

   2. Taylor expanded in y around inf 70.2%

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

   if 2.14999999999999996e77 < t

   1. Initial program 84.7%

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

   2. Taylor expanded in y around 0 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   4. Simplified 69.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 6: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+24} \lor \neg \left(t \leq 6.5 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.9e+24) (not (<= t 6.5e+216))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.9e+24) || !(t <= 6.5e+216)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: 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 ((t <= (-4.9d+24)) .or. (.not. (t <= 6.5d+216))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.9e+24) || !(t <= 6.5e+216)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -4.9e+24) or not (t <= 6.5e+216):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.9e+24) || !(t <= 6.5e+216))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.9e+24) || ~((t <= 6.5e+216)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.9e+24], N[Not[LessEqual[t, 6.5e+216]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.90000000000000029e24 or 6.50000000000000029e216 < t

   1. Initial program 91.7%

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

      1. clear-num 90.2%

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

      2. associate-/r/ 91.6%

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

   3. Applied egg-rr 91.6%

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

   4. Taylor expanded in y around 0 70.6%

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

      1. associate-/r* 70.5%

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

   6. Simplified 70.5%

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

      1. associate-*l/ 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. expm1-log1p-u 70.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1}{t} \cdot x}{z}\right)\right)} \]

      2. expm1-udef 44.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-1}{t} \cdot x}{z}\right)} - 1} \]

      3. *-commutative 44.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x \cdot \frac{-1}{t}}}{z}\right)} - 1 \]

      4. frac-2neg 44.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot \color{blue}{\frac{--1}{-t}}}{z}\right)} - 1 \]

      5. metadata-eval 44.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot \frac{\color{blue}{1}}{-t}}{z}\right)} - 1 \]

      6. un-div-inv 44.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{-t}}}{z}\right)} - 1 \]

      7. add-sqr-sqrt 36.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{x}{\color{blue}{\sqrt{-t} \cdot \sqrt{-t}}}}{z}\right)} - 1 \]

      8. sqrt-unprod 42.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{x}{\color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}}}{z}\right)} - 1 \]

      9. sqr-neg 42.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{x}{\sqrt{\color{blue}{t \cdot t}}}}{z}\right)} - 1 \]

      10. sqrt-unprod 6.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{x}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}}{z}\right)} - 1 \]

      11. add-sqr-sqrt 39.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{x}{\color{blue}{t}}}{z}\right)} - 1 \]

   10. Applied egg-rr 39.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 39.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

       2. expm1-log1p 39.3%

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

       3. associate-/r* 36.3%

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

   12. Simplified 36.3%

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

   if -4.90000000000000029e24 < t < 6.50000000000000029e216

   1. Initial program 96.1%

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

   2. Taylor expanded in y around inf 60.4%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+24} \lor \neg \left(t \leq 6.5 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 54.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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 94.7%

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

2. Taylor expanded in y around inf 50.6%

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

3. Final simplification 50.6%

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

Developer target: 96.5% accurate, 0.5× 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 2023185 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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