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

Percentage Accurate: 95.6% → 95.6%

Time: 7.9s

Alternatives: 6

Speedup: 1.0×

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: 95.6% 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: 95.6% 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]

Derivation

1. Initial program 96.8%

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

Alternative 2: 72.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.48 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-213} \lor \neg \left(y \leq -3.5 \cdot 10^{-299} \lor \neg \left(y \leq -3.3 \cdot 10^{-299}\right) \land y \leq 4.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.9e-45)
   (/ x y)
   (if (<= y -7e-84)
     (/ (- x) (* z t))
     (if (<= y -1.48e-102)
       (/ (/ x t) (- z))
       (if (<= y -5.8e-183)
         (/ (/ x z) (- t))
         (if (or (<= y -6.2e-213)
                 (not
                  (or (<= y -3.5e-299)
                      (and (not (<= y -3.3e-299)) (<= y 4.1e-45)))))
           (/ x y)
           (* x (/ (/ -1.0 t) z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.9e-45) {
		tmp = x / y;
	} else if (y <= -7e-84) {
		tmp = -x / (z * t);
	} else if (y <= -1.48e-102) {
		tmp = (x / t) / -z;
	} else if (y <= -5.8e-183) {
		tmp = (x / z) / -t;
	} else if ((y <= -6.2e-213) || !((y <= -3.5e-299) || (!(y <= -3.3e-299) && (y <= 4.1e-45)))) {
		tmp = x / y;
	} else {
		tmp = x * ((-1.0 / t) / z);
	}
	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) :: tmp
    if (y <= (-4.9d-45)) then
        tmp = x / y
    else if (y <= (-7d-84)) then
        tmp = -x / (z * t)
    else if (y <= (-1.48d-102)) then
        tmp = (x / t) / -z
    else if (y <= (-5.8d-183)) then
        tmp = (x / z) / -t
    else if ((y <= (-6.2d-213)) .or. (.not. (y <= (-3.5d-299)) .or. (.not. (y <= (-3.3d-299))) .and. (y <= 4.1d-45))) then
        tmp = x / y
    else
        tmp = x * (((-1.0d0) / t) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.9e-45) {
		tmp = x / y;
	} else if (y <= -7e-84) {
		tmp = -x / (z * t);
	} else if (y <= -1.48e-102) {
		tmp = (x / t) / -z;
	} else if (y <= -5.8e-183) {
		tmp = (x / z) / -t;
	} else if ((y <= -6.2e-213) || !((y <= -3.5e-299) || (!(y <= -3.3e-299) && (y <= 4.1e-45)))) {
		tmp = x / y;
	} else {
		tmp = x * ((-1.0 / t) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.9e-45:
		tmp = x / y
	elif y <= -7e-84:
		tmp = -x / (z * t)
	elif y <= -1.48e-102:
		tmp = (x / t) / -z
	elif y <= -5.8e-183:
		tmp = (x / z) / -t
	elif (y <= -6.2e-213) or not ((y <= -3.5e-299) or (not (y <= -3.3e-299) and (y <= 4.1e-45))):
		tmp = x / y
	else:
		tmp = x * ((-1.0 / t) / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.9e-45)
		tmp = Float64(x / y);
	elseif (y <= -7e-84)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (y <= -1.48e-102)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (y <= -5.8e-183)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif ((y <= -6.2e-213) || !((y <= -3.5e-299) || (!(y <= -3.3e-299) && (y <= 4.1e-45))))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(Float64(-1.0 / t) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.9e-45)
		tmp = x / y;
	elseif (y <= -7e-84)
		tmp = -x / (z * t);
	elseif (y <= -1.48e-102)
		tmp = (x / t) / -z;
	elseif (y <= -5.8e-183)
		tmp = (x / z) / -t;
	elseif ((y <= -6.2e-213) || ~(((y <= -3.5e-299) || (~((y <= -3.3e-299)) && (y <= 4.1e-45)))))
		tmp = x / y;
	else
		tmp = x * ((-1.0 / t) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.9e-45], N[(x / y), $MachinePrecision], If[LessEqual[y, -7e-84], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.48e-102], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[y, -5.8e-183], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[Or[LessEqual[y, -6.2e-213], N[Not[Or[LessEqual[y, -3.5e-299], And[N[Not[LessEqual[y, -3.3e-299]], $MachinePrecision], LessEqual[y, 4.1e-45]]]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(N[(-1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.8999999999999998e-45 or -5.8000000000000001e-183 < y < -6.1999999999999996e-213 or -3.49999999999999991e-299 < y < -3.3000000000000002e-299 or 4.0999999999999999e-45 < y

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf 80.7%

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

   if -4.8999999999999998e-45 < y < -7.0000000000000002e-84

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. neg-mul-1 78.8%

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

   5. Simplified 78.8%

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

   if -7.0000000000000002e-84 < y < -1.48e-102

   1. Initial program 75.8%

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

      1. clear-num 76.2%

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

      2. associate-/r/ 76.2%

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

   4. Applied egg-rr 76.2%

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

   5. Taylor expanded in y around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. associate-/r* 75.8%

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

      3. distribute-neg-frac2 75.8%

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

   7. Simplified 75.8%

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

   if -1.48e-102 < y < -5.8000000000000001e-183

   1. Initial program 93.5%

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

   3. Taylor expanded in t around -inf 78.6%

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

   4. Taylor expanded in z around inf 81.4%

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

   if -6.1999999999999996e-213 < y < -3.49999999999999991e-299 or -3.3000000000000002e-299 < y < 4.0999999999999999e-45

   1. Initial program 96.9%

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

      1. clear-num 96.7%

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

      2. associate-/r/ 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in y around 0 78.7%

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

      1. associate-/r* 78.8%

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

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.48 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-213} \lor \neg \left(y \leq -3.5 \cdot 10^{-299} \lor \neg \left(y \leq -3.3 \cdot 10^{-299}\right) \land y \leq 4.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-299} \lor \neg \left(y \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x z) (- t))) (t_2 (/ (- x) (* z t))))
   (if (<= y -1.26e-46)
     (/ x y)
     (if (<= y -1.3e-86)
       t_2
       (if (<= y -3.6e-102)
         (/ (/ x t) (- z))
         (if (<= y -5.8e-183)
           t_1
           (if (<= y -6.2e-213)
             (/ x y)
             (if (<= y -3.5e-299)
               t_1
               (if (or (<= y -3.3e-299) (not (<= y 8e-49)))
                 (/ x y)
                 t_2)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double t_2 = -x / (z * t);
	double tmp;
	if (y <= -1.26e-46) {
		tmp = x / y;
	} else if (y <= -1.3e-86) {
		tmp = t_2;
	} else if (y <= -3.6e-102) {
		tmp = (x / t) / -z;
	} else if (y <= -5.8e-183) {
		tmp = t_1;
	} else if (y <= -6.2e-213) {
		tmp = x / y;
	} else if (y <= -3.5e-299) {
		tmp = t_1;
	} else if ((y <= -3.3e-299) || !(y <= 8e-49)) {
		tmp = x / y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (x / z) / -t
    t_2 = -x / (z * t)
    if (y <= (-1.26d-46)) then
        tmp = x / y
    else if (y <= (-1.3d-86)) then
        tmp = t_2
    else if (y <= (-3.6d-102)) then
        tmp = (x / t) / -z
    else if (y <= (-5.8d-183)) then
        tmp = t_1
    else if (y <= (-6.2d-213)) then
        tmp = x / y
    else if (y <= (-3.5d-299)) then
        tmp = t_1
    else if ((y <= (-3.3d-299)) .or. (.not. (y <= 8d-49))) then
        tmp = x / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double t_2 = -x / (z * t);
	double tmp;
	if (y <= -1.26e-46) {
		tmp = x / y;
	} else if (y <= -1.3e-86) {
		tmp = t_2;
	} else if (y <= -3.6e-102) {
		tmp = (x / t) / -z;
	} else if (y <= -5.8e-183) {
		tmp = t_1;
	} else if (y <= -6.2e-213) {
		tmp = x / y;
	} else if (y <= -3.5e-299) {
		tmp = t_1;
	} else if ((y <= -3.3e-299) || !(y <= 8e-49)) {
		tmp = x / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) / -t
	t_2 = -x / (z * t)
	tmp = 0
	if y <= -1.26e-46:
		tmp = x / y
	elif y <= -1.3e-86:
		tmp = t_2
	elif y <= -3.6e-102:
		tmp = (x / t) / -z
	elif y <= -5.8e-183:
		tmp = t_1
	elif y <= -6.2e-213:
		tmp = x / y
	elif y <= -3.5e-299:
		tmp = t_1
	elif (y <= -3.3e-299) or not (y <= 8e-49):
		tmp = x / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) / Float64(-t))
	t_2 = Float64(Float64(-x) / Float64(z * t))
	tmp = 0.0
	if (y <= -1.26e-46)
		tmp = Float64(x / y);
	elseif (y <= -1.3e-86)
		tmp = t_2;
	elseif (y <= -3.6e-102)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (y <= -5.8e-183)
		tmp = t_1;
	elseif (y <= -6.2e-213)
		tmp = Float64(x / y);
	elseif (y <= -3.5e-299)
		tmp = t_1;
	elseif ((y <= -3.3e-299) || !(y <= 8e-49))
		tmp = Float64(x / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) / -t;
	t_2 = -x / (z * t);
	tmp = 0.0;
	if (y <= -1.26e-46)
		tmp = x / y;
	elseif (y <= -1.3e-86)
		tmp = t_2;
	elseif (y <= -3.6e-102)
		tmp = (x / t) / -z;
	elseif (y <= -5.8e-183)
		tmp = t_1;
	elseif (y <= -6.2e-213)
		tmp = x / y;
	elseif (y <= -3.5e-299)
		tmp = t_1;
	elseif ((y <= -3.3e-299) || ~((y <= 8e-49)))
		tmp = x / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.26e-46], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.3e-86], t$95$2, If[LessEqual[y, -3.6e-102], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[y, -5.8e-183], t$95$1, If[LessEqual[y, -6.2e-213], N[(x / y), $MachinePrecision], If[LessEqual[y, -3.5e-299], t$95$1, If[Or[LessEqual[y, -3.3e-299], N[Not[LessEqual[y, 8e-49]], $MachinePrecision]], N[(x / y), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.26000000000000004e-46 or -5.8000000000000001e-183 < y < -6.1999999999999996e-213 or -3.49999999999999991e-299 < y < -3.3000000000000002e-299 or 7.99999999999999949e-49 < y

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf 80.3%

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

   if -1.26000000000000004e-46 < y < -1.3000000000000001e-86 or -3.3000000000000002e-299 < y < 7.99999999999999949e-49

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

   5. Simplified 76.2%

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

   if -1.3000000000000001e-86 < y < -3.6e-102

   1. Initial program 75.8%

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

      1. clear-num 76.2%

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

      2. associate-/r/ 76.2%

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

   4. Applied egg-rr 76.2%

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

   5. Taylor expanded in y around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. associate-/r* 75.8%

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

      3. distribute-neg-frac2 75.8%

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

   7. Simplified 75.8%

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

   if -3.6e-102 < y < -5.8000000000000001e-183 or -6.1999999999999996e-213 < y < -3.49999999999999991e-299

   1. Initial program 96.2%

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

   3. Taylor expanded in t around -inf 66.8%

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

   4. Taylor expanded in z around inf 75.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-299} \lor \neg \left(y \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ t_2 := \frac{\frac{x}{t}}{-z}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+224}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x) (* z t))) (t_2 (/ (/ x t) (- z))))
   (if (<= z -1.7e+224)
     t_2
     (if (<= z -1.45e+130)
       t_1
       (if (<= z -7.5e+45)
         (/ x y)
         (if (<= z -1.6e+31) t_2 (if (<= z 5.5e-122) (/ x y) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double t_2 = (x / t) / -z;
	double tmp;
	if (z <= -1.7e+224) {
		tmp = t_2;
	} else if (z <= -1.45e+130) {
		tmp = t_1;
	} else if (z <= -7.5e+45) {
		tmp = x / y;
	} else if (z <= -1.6e+31) {
		tmp = t_2;
	} else if (z <= 5.5e-122) {
		tmp = x / y;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = -x / (z * t)
    t_2 = (x / t) / -z
    if (z <= (-1.7d+224)) then
        tmp = t_2
    else if (z <= (-1.45d+130)) then
        tmp = t_1
    else if (z <= (-7.5d+45)) then
        tmp = x / y
    else if (z <= (-1.6d+31)) then
        tmp = t_2
    else if (z <= 5.5d-122) then
        tmp = x / y
    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 = -x / (z * t);
	double t_2 = (x / t) / -z;
	double tmp;
	if (z <= -1.7e+224) {
		tmp = t_2;
	} else if (z <= -1.45e+130) {
		tmp = t_1;
	} else if (z <= -7.5e+45) {
		tmp = x / y;
	} else if (z <= -1.6e+31) {
		tmp = t_2;
	} else if (z <= 5.5e-122) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -x / (z * t)
	t_2 = (x / t) / -z
	tmp = 0
	if z <= -1.7e+224:
		tmp = t_2
	elif z <= -1.45e+130:
		tmp = t_1
	elif z <= -7.5e+45:
		tmp = x / y
	elif z <= -1.6e+31:
		tmp = t_2
	elif z <= 5.5e-122:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	t_2 = Float64(Float64(x / t) / Float64(-z))
	tmp = 0.0
	if (z <= -1.7e+224)
		tmp = t_2;
	elseif (z <= -1.45e+130)
		tmp = t_1;
	elseif (z <= -7.5e+45)
		tmp = Float64(x / y);
	elseif (z <= -1.6e+31)
		tmp = t_2;
	elseif (z <= 5.5e-122)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -x / (z * t);
	t_2 = (x / t) / -z;
	tmp = 0.0;
	if (z <= -1.7e+224)
		tmp = t_2;
	elseif (z <= -1.45e+130)
		tmp = t_1;
	elseif (z <= -7.5e+45)
		tmp = x / y;
	elseif (z <= -1.6e+31)
		tmp = t_2;
	elseif (z <= 5.5e-122)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]}, If[LessEqual[z, -1.7e+224], t$95$2, If[LessEqual[z, -1.45e+130], t$95$1, If[LessEqual[z, -7.5e+45], N[(x / y), $MachinePrecision], If[LessEqual[z, -1.6e+31], t$95$2, If[LessEqual[z, 5.5e-122], N[(x / y), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7000000000000001e224 or -7.50000000000000058e45 < z < -1.6e31

   1. Initial program 86.6%

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

      1. clear-num 86.4%

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

      2. associate-/r/ 86.6%

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

   4. Applied egg-rr 86.6%

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

   5. Taylor expanded in y around 0 69.4%

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

      1. mul-1-neg 69.4%

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

      2. associate-/r* 82.4%

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

      3. distribute-neg-frac2 82.4%

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

   7. Simplified 82.4%

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

   if -1.7000000000000001e224 < z < -1.45e130 or 5.50000000000000053e-122 < z

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0 59.2%

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

      1. associate-*r/ 59.2%

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

      2. neg-mul-1 59.2%

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

   5. Simplified 59.2%

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

   if -1.45e130 < z < -7.50000000000000058e45 or -1.6e31 < z < 5.50000000000000053e-122

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf 74.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+224}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-41} \lor \neg \left(y \leq -1.2 \cdot 10^{-150}\right) \land \left(y \leq -6.2 \cdot 10^{-213} \lor \neg \left(y \leq 5.2 \cdot 10^{-67}\right)\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.95e-41)
         (and (not (<= y -1.2e-150))
              (or (<= y -6.2e-213) (not (<= y 5.2e-67)))))
   (/ x y)
   (/ (- x) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.95e-41) || (!(y <= -1.2e-150) && ((y <= -6.2e-213) || !(y <= 5.2e-67)))) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	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) :: tmp
    if ((y <= (-1.95d-41)) .or. (.not. (y <= (-1.2d-150))) .and. (y <= (-6.2d-213)) .or. (.not. (y <= 5.2d-67))) then
        tmp = x / y
    else
        tmp = -x / (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.95e-41) || (!(y <= -1.2e-150) && ((y <= -6.2e-213) || !(y <= 5.2e-67)))) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.95e-41) or (not (y <= -1.2e-150) and ((y <= -6.2e-213) or not (y <= 5.2e-67))):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.95e-41) || (!(y <= -1.2e-150) && ((y <= -6.2e-213) || !(y <= 5.2e-67))))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-x) / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.95e-41) || (~((y <= -1.2e-150)) && ((y <= -6.2e-213) || ~((y <= 5.2e-67)))))
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.95e-41], And[N[Not[LessEqual[y, -1.2e-150]], $MachinePrecision], Or[LessEqual[y, -6.2e-213], N[Not[LessEqual[y, 5.2e-67]], $MachinePrecision]]]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999995e-41 or -1.2e-150 < y < -6.1999999999999996e-213 or 5.1999999999999998e-67 < y

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf 77.5%

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

   if -1.94999999999999995e-41 < y < -1.2e-150 or -6.1999999999999996e-213 < y < 5.1999999999999998e-67

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 78.5%

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

      1. associate-*r/ 78.5%

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

      2. neg-mul-1 78.5%

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

   5. Simplified 78.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-41} \lor \neg \left(y \leq -1.2 \cdot 10^{-150}\right) \land \left(y \leq -6.2 \cdot 10^{-213} \lor \neg \left(y \leq 5.2 \cdot 10^{-67}\right)\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in y around inf 58.4%

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

Developer target: 96.5% 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 2024103 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (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))))