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

Percentage Accurate: 95.7% → 97.9%

Time: 9.9s

Alternatives: 5

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: 95.7% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

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 (<= (* z t) 2e+306) (/ x (- y (* z t))) (/ (/ x 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) <= 2e+306) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) / -t;
	}
	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+306) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / z) / -t
    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+306) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / 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) <= 2e+306:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / 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) <= 2e+306)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-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) <= 2e+306)
		tmp = x / (y - (z * t));
	else
		tmp = (x / 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[LessEqual[N[(z * t), $MachinePrecision], 2e+306], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2.00000000000000003e306

   1. Initial program 99.0%

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

   if 2.00000000000000003e306 < (*.f64 z t)

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

      1. distribute-frac-neg 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-/r* 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-9} \lor \neg \left(z \leq 7 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

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 -2.1e-9) (not (<= z 7e-83))) (/ (/ x z) (- t)) (/ 1.0 (/ y x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.1e-9) || !(z <= 7e-83)) {
		tmp = (x / z) / -t;
	} else {
		tmp = 1.0 / (y / x);
	}
	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 <= (-2.1d-9)) .or. (.not. (z <= 7d-83))) then
        tmp = (x / z) / -t
    else
        tmp = 1.0d0 / (y / x)
    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 <= -2.1e-9) || !(z <= 7e-83)) {
		tmp = (x / z) / -t;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -2.1e-9) or not (z <= 7e-83):
		tmp = (x / z) / -t
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.1e-9) || !(z <= 7e-83))
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(1.0 / Float64(y / x));
	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 <= -2.1e-9) || ~((z <= 7e-83)))
		tmp = (x / z) / -t;
	else
		tmp = 1.0 / (y / x);
	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[z, -2.1e-9], N[Not[LessEqual[z, 7e-83]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000019e-9 or 7.00000000000000061e-83 < z

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 0 70.8%

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

      1. associate-*r/ 70.8%

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

      2. neg-mul-1 70.8%

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

   5. Simplified 70.8%

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

      1. distribute-frac-neg 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-/r* 72.0%

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

   7. Applied egg-rr 72.0%

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

   if -2.10000000000000019e-9 < z < 7.00000000000000061e-83

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 73.3%

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

      1. clear-num 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. inv-pow 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. unpow-1 72.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   7. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-9} \lor \neg \left(z \leq 7 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-53} \lor \neg \left(t \leq 3.4 \cdot 10^{+195}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= t -1.55e-53) (not (<= t 3.4e+195))) (/ 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 ((t <= -1.55e-53) || !(t <= 3.4e+195)) {
		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 ((t <= (-1.55d-53)) .or. (.not. (t <= 3.4d+195))) 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 ((t <= -1.55e-53) || !(t <= 3.4e+195)) {
		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 (t <= -1.55e-53) or not (t <= 3.4e+195):
		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 ((t <= -1.55e-53) || !(t <= 3.4e+195))
		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 ((t <= -1.55e-53) || ~((t <= 3.4e+195)))
		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[t, -1.55e-53], N[Not[LessEqual[t, 3.4e+195]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55000000000000008e-53 or 3.4000000000000001e195 < t

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

   5. Simplified 78.7%

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

      1. neg-sub0 78.7%

         \[\leadsto \frac{\color{blue}{0 - x}}{t \cdot z} \]

      2. sub-neg 78.7%

         \[\leadsto \frac{\color{blue}{0 + \left(-x\right)}}{t \cdot z} \]

      3. add-sqr-sqrt 42.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z} \]

      4. sqrt-unprod 51.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z} \]

      5. sqr-neg 51.6%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{x \cdot x}}}{t \cdot z} \]

      6. sqrt-unprod 15.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z} \]

      7. add-sqr-sqrt 40.8%

         \[\leadsto \frac{0 + \color{blue}{x}}{t \cdot z} \]

   7. Applied egg-rr 40.8%

      \[\leadsto \frac{\color{blue}{0 + x}}{t \cdot z} \]
   8. Step-by-step derivation

      1. +-lft-identity 40.8%

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

   9. Simplified 40.8%

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

   if -1.55000000000000008e-53 < t < 3.4000000000000001e195

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf 60.6%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-53} \lor \neg \left(t \leq 3.4 \cdot 10^{+195}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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 (<= t -1.55e-53)
   (/ x (* z t))
   (if (<= t 1.3e+197) (/ 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 (t <= -1.55e-53) {
		tmp = x / (z * t);
	} else if (t <= 1.3e+197) {
		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 (t <= (-1.55d-53)) then
        tmp = x / (z * t)
    else if (t <= 1.3d+197) 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 (t <= -1.55e-53) {
		tmp = x / (z * t);
	} else if (t <= 1.3e+197) {
		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 t <= -1.55e-53:
		tmp = x / (z * t)
	elif t <= 1.3e+197:
		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 (t <= -1.55e-53)
		tmp = Float64(x / Float64(z * t));
	elseif (t <= 1.3e+197)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / 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 (t <= -1.55e-53)
		tmp = x / (z * t);
	elseif (t <= 1.3e+197)
		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[t, -1.55e-53], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e+197], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.55000000000000008e-53

   1. Initial program 93.9%

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

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

      1. neg-sub0 76.2%

         \[\leadsto \frac{\color{blue}{0 - x}}{t \cdot z} \]

      2. sub-neg 76.2%

         \[\leadsto \frac{\color{blue}{0 + \left(-x\right)}}{t \cdot z} \]

      3. add-sqr-sqrt 37.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z} \]

      4. sqrt-unprod 48.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z} \]

      5. sqr-neg 48.4%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{x \cdot x}}}{t \cdot z} \]

      6. sqrt-unprod 17.7%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z} \]

      7. add-sqr-sqrt 35.5%

         \[\leadsto \frac{0 + \color{blue}{x}}{t \cdot z} \]

   7. Applied egg-rr 35.5%

      \[\leadsto \frac{\color{blue}{0 + x}}{t \cdot z} \]
   8. Step-by-step derivation

      1. +-lft-identity 35.5%

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

   9. Simplified 35.5%

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

   if -1.55000000000000008e-53 < t < 1.29999999999999994e197

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf 60.6%

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

   if 1.29999999999999994e197 < t

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   5. Simplified 86.5%

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

      1. add-sqr-sqrt 57.4%

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

      2. sqrt-unprod 61.6%

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

      3. sqr-neg 61.6%

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

      4. sqrt-unprod 10.4%

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

      5. add-sqr-sqrt 57.2%

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

      6. div-inv 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. associate-*r/ 57.2%

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

      2. *-rgt-identity 57.2%

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

      3. associate-/r* 57.2%

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

   9. Simplified 57.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% 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 97.2%

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

3. Taylor expanded in y around inf 48.3%

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

Developer target: 96.9% 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 2024110 
(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))))