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

Percentage Accurate: 95.9% → 99.8%

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: 95.9% 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, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+251}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \end{array} \]

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) 1e+251)))
   (/ (/ x z) (- t))
   (/ x (fma z (- t) y))))

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) <= 1e+251)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / fma(z, -t, 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) <= Float64(-Inf)) || !(Float64(z * t) <= 1e+251))
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / fma(z, Float64(-t), y));
	end
	return 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], 1e+251]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1e251 < (*.f64 z t)

   1. Initial program 72.2%

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

   3. Taylor expanded in z around inf 85.9%

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

      1. distribute-lft-out 85.9%

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

      2. associate-*r/ 85.9%

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

      3. mul-1-neg 85.9%

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

      4. *-commutative 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in t around inf 99.9%

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

   7. Taylor expanded in x around 0 72.2%

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

      1. associate-/l/ 99.8%

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

   9. Simplified 99.8%

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

   if -inf.0 < (*.f64 z t) < 1e251

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, -t, y\right)}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+251}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+251}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

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) 1e+251)))
   (/ (/ 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) <= 1e+251)) {
		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) <= 1e+251)) {
		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) <= 1e+251):
		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) <= 1e+251))
		tmp = Float64(Float64(x / z) / Float64(-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) <= 1e+251)))
		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], 1e+251]], $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 1e251 < (*.f64 z t)

   1. Initial program 72.2%

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

   3. Taylor expanded in z around inf 85.9%

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

      1. distribute-lft-out 85.9%

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

      2. associate-*r/ 85.9%

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

      3. mul-1-neg 85.9%

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

      4. *-commutative 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in t around inf 99.9%

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

   7. Taylor expanded in x around 0 72.2%

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

      1. associate-/l/ 99.8%

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

   9. Simplified 99.8%

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

   if -inf.0 < (*.f64 z t) < 1e251

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+251}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.0002 \lor \neg \left(z \cdot t \leq 0.2\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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) -0.0002) (not (<= (* z t) 0.2)))
   (/ (/ x t) (- z))
   (/ 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) <= -0.0002) || !((z * t) <= 0.2)) {
		tmp = (x / t) / -z;
	} 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) <= (-0.0002d0)) .or. (.not. ((z * t) <= 0.2d0))) then
        tmp = (x / t) / -z
    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) <= -0.0002) || !((z * t) <= 0.2)) {
		tmp = (x / t) / -z;
	} 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) <= -0.0002) or not ((z * t) <= 0.2):
		tmp = (x / t) / -z
	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) <= -0.0002) || !(Float64(z * t) <= 0.2))
		tmp = Float64(Float64(x / t) / Float64(-z));
	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) <= -0.0002) || ~(((z * t) <= 0.2)))
		tmp = (x / t) / -z;
	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], -0.0002], N[Not[LessEqual[N[(z * t), $MachinePrecision], 0.2]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e-4 or 0.20000000000000001 < (*.f64 z t)

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf 74.3%

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

      1. distribute-lft-out 74.3%

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

      2. associate-*r/ 74.3%

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

      3. mul-1-neg 74.3%

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

      4. *-commutative 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in t around inf 85.9%

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

   if -2.0000000000000001e-4 < (*.f64 z t) < 0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 85.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.0002 \lor \neg \left(z \cdot t \leq 0.2\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.0002:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq 0.2:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

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 (<= (* z t) -0.0002)
   (/ -1.0 (* z (/ t x)))
   (if (<= (* z t) 0.2) (/ x y) (/ (/ x t) (- z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -0.0002) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= 0.2) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	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) <= (-0.0002d0)) then
        tmp = (-1.0d0) / (z * (t / x))
    else if ((z * t) <= 0.2d0) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    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) <= -0.0002) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= 0.2) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -0.0002:
		tmp = -1.0 / (z * (t / x))
	elif (z * t) <= 0.2:
		tmp = x / y
	else:
		tmp = (x / t) / -z
	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) <= -0.0002)
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	elseif (Float64(z * t) <= 0.2)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	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) <= -0.0002)
		tmp = -1.0 / (z * (t / x));
	elseif ((z * t) <= 0.2)
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	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[LessEqual[N[(z * t), $MachinePrecision], -0.0002], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.2], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.0000000000000001e-4

   1. Initial program 92.3%

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

   3. Taylor expanded in z around inf 74.9%

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

      1. distribute-lft-out 74.9%

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

      2. associate-*r/ 74.9%

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

      3. mul-1-neg 74.9%

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

      4. *-commutative 74.9%

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

   5. Simplified 74.9%

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

   6. Taylor expanded in t around inf 85.9%

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

      1. neg-mul-1 85.9%

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

      2. clear-num 85.9%

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

      3. un-div-inv 85.9%

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

      4. div-inv 87.4%

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

      5. clear-num 87.4%

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

   8. Applied egg-rr 87.4%

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

   if -2.0000000000000001e-4 < (*.f64 z t) < 0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 85.4%

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

   if 0.20000000000000001 < (*.f64 z t)

   1. Initial program 88.9%

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

   3. Taylor expanded in z around inf 73.7%

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

      1. distribute-lft-out 73.7%

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

      2. associate-*r/ 73.7%

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

      3. mul-1-neg 73.7%

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

      4. *-commutative 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in t around inf 85.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.0002:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq 0.2:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+137} \lor \neg \left(z \cdot t \leq 10^{+159}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+137) (not (<= (* z t) 1e+159)))
   (/ 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+137) || !((z * t) <= 1e+159)) {
		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+137)) .or. (.not. ((z * t) <= 1d+159))) 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+137) || !((z * t) <= 1e+159)) {
		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+137) or not ((z * t) <= 1e+159):
		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+137) || !(Float64(z * t) <= 1e+159))
		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+137) || ~(((z * t) <= 1e+159)))
		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+137], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+159]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e137 or 9.9999999999999993e158 < (*.f64 z t)

   1. Initial program 83.8%

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

   3. Taylor expanded in z around inf 86.8%

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

      1. distribute-lft-out 86.8%

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

      2. associate-*r/ 86.8%

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

      3. mul-1-neg 86.8%

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

      4. *-commutative 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in t around inf 99.9%

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

      1. add-sqr-sqrt 78.3%

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

      2. sqrt-unprod 67.1%

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

      3. sqr-neg 67.1%

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

      4. sqrt-unprod 60.3%

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

      5. add-sqr-sqrt 61.4%

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

      6. associate-/l/ 61.7%

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

   8. Applied egg-rr 61.7%

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

   if -2.0000000000000001e137 < (*.f64 z t) < 9.9999999999999993e158

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 69.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+137} \lor \neg \left(z \cdot t \leq 10^{+159}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.0% 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 95.2%

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

3. Taylor expanded in y around inf 54.3%

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

Developer Target 1: 96.3% 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 2024180 
(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))))