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

Percentage Accurate: 96.1% → 97.6%

Time: 12.2s

Alternatives: 8

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: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+239) (/ (/ -1.0 t) (/ z x)) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+239) {
		tmp = (-1.0 / t) / (z / x);
	} else {
		tmp = x / (y - (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 ((z * t) <= (-1d+239)) then
        tmp = ((-1.0d0) / t) / (z / x)
    else
        tmp = x / (y - (z * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+239) {
		tmp = (-1.0 / t) / (z / x);
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e+239:
		tmp = (-1.0 / t) / (z / x)
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+239)
		tmp = Float64(Float64(-1.0 / t) / Float64(z / x));
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -1e+239)
		tmp = (-1.0 / t) / (z / x);
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+239], N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999991e238

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. *-commutative 74.6%

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

      3. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -9.99999999999999991e238 < (*.f64 z t)

   1. Initial program 98.2%

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

Alternative 2: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e+44) (not (<= (* z t) 2e-33)))
   (/ (/ x z) (- t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+44) || !((z * t) <= 2e-33)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / y;
	}
	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 (((z * t) <= (-1d+44)) .or. (.not. ((z * t) <= 2d-33))) then
        tmp = (x / z) / -t
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+44) || !((z * t) <= 2e-33)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e+44) or not ((z * t) <= 2e-33):
		tmp = (x / z) / -t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+44) || !(Float64(z * t) <= 2e-33))
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e+44) || ~(((z * t) <= 2e-33)))
		tmp = (x / z) / -t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+44], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-33]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.0000000000000001e44 or 2.0000000000000001e-33 < (*.f64 z t)

   1. Initial program 91.4%

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

   3. Taylor expanded in t around inf 75.2%

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

      1. distribute-lft-out 75.2%

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

      2. associate-*r/ 75.2%

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

      3. mul-1-neg 75.2%

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

      4. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in z around inf 83.9%

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

   if -1.0000000000000001e44 < (*.f64 z t) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -4e+21)
   (/ -1.0 (* z (/ t x)))
   (if (<= (* z t) 2e-33) (/ x y) (/ x (- (* z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -4e+21) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= 2e-33) {
		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 ((z * t) <= (-4d+21)) then
        tmp = (-1.0d0) / (z * (t / x))
    else if ((z * t) <= 2d-33) 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 ((z * t) <= -4e+21) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= 2e-33) {
		tmp = x / y;
	} else {
		tmp = x / -(z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -4e+21:
		tmp = -1.0 / (z * (t / x))
	elif (z * t) <= 2e-33:
		tmp = x / y
	else:
		tmp = x / -(z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -4e+21)
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	elseif (Float64(z * t) <= 2e-33)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(-Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -4e+21)
		tmp = -1.0 / (z * (t / x));
	elseif ((z * t) <= 2e-33)
		tmp = x / y;
	else
		tmp = x / -(z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -4e+21], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-33], N[(x / y), $MachinePrecision], N[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4e21

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. *-commutative 85.3%

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

      3. times-frac 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in y around 0 87.4%

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

      1. clear-num 87.3%

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

      2. frac-times 87.8%

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

      3. metadata-eval 87.8%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-/l* 86.5%

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

   8. Applied egg-rr 86.5%

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

   if -4e21 < (*.f64 z t) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.5%

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

   if 2.0000000000000001e-33 < (*.f64 z t)

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. neg-mul-1 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -4e+21)
   (/ (/ (- x) t) z)
   (if (<= (* z t) 2e-33) (/ x y) (/ x (- (* z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -4e+21) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 2e-33) {
		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 ((z * t) <= (-4d+21)) then
        tmp = (-x / t) / z
    else if ((z * t) <= 2d-33) 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 ((z * t) <= -4e+21) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 2e-33) {
		tmp = x / y;
	} else {
		tmp = x / -(z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -4e+21:
		tmp = (-x / t) / z
	elif (z * t) <= 2e-33:
		tmp = x / y
	else:
		tmp = x / -(z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -4e+21)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 2e-33)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(-Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -4e+21)
		tmp = (-x / t) / z;
	elseif ((z * t) <= 2e-33)
		tmp = x / y;
	else
		tmp = x / -(z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -4e+21], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-33], N[(x / y), $MachinePrecision], N[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4e21

   1. Initial program 88.2%

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

   3. Taylor expanded in y around inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in y around 0 76.8%

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

      1. associate-/r* 86.0%

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

      2. associate-*r/ 86.0%

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

      3. associate-*r/ 86.0%

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

      4. neg-mul-1 86.0%

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

   8. Simplified 86.0%

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

   if -4e21 < (*.f64 z t) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 86.5%

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

   if 2.0000000000000001e-33 < (*.f64 z t)

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. neg-mul-1 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+44)
   (/ (/ x z) (- t))
   (if (<= (* z t) 2e-33) (/ x y) (/ x (- (* z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+44) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e-33) {
		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 ((z * t) <= (-1d+44)) then
        tmp = (x / z) / -t
    else if ((z * t) <= 2d-33) 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 ((z * t) <= -1e+44) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e-33) {
		tmp = x / y;
	} else {
		tmp = x / -(z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e+44:
		tmp = (x / z) / -t
	elif (z * t) <= 2e-33:
		tmp = x / y
	else:
		tmp = x / -(z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+44)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (Float64(z * t) <= 2e-33)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(-Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -1e+44)
		tmp = (x / z) / -t;
	elseif ((z * t) <= 2e-33)
		tmp = x / y;
	else
		tmp = x / -(z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+44], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-33], N[(x / y), $MachinePrecision], N[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.0000000000000001e44

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf 80.5%

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

      1. distribute-lft-out 80.5%

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

      2. associate-*r/ 80.5%

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

      3. mul-1-neg 80.5%

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

      4. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in z around inf 89.7%

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

   if -1.0000000000000001e44 < (*.f64 z t) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 85.3%

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

   if 2.0000000000000001e-33 < (*.f64 z t)

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. neg-mul-1 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+231} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+231) (not (<= (* z t) 4e+206)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+231) || !((z * t) <= 4e+206)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	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 (((z * t) <= (-5d+231)) .or. (.not. ((z * t) <= 4d+206))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+231) || !((z * t) <= 4e+206)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+231) or not ((z * t) <= 4e+206):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+231) || !(Float64(z * t) <= 4e+206))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+231) || ~(((z * t) <= 4e+206)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+231], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e+206]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000028e231 or 4.0000000000000002e206 < (*.f64 z t)

   1. Initial program 81.3%

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

   3. Taylor expanded in t around inf 86.8%

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

      1. distribute-lft-out 86.8%

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

      2. associate-*r/ 86.8%

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

      3. mul-1-neg 86.8%

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

      4. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 99.9%

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

      1. add-sqr-sqrt 76.9%

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

      2. sqrt-unprod 66.5%

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

      3. sqr-neg 66.5%

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

      4. sqrt-unprod 53.4%

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

      5. add-sqr-sqrt 54.3%

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

      6. div-inv 54.3%

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

      7. frac-times 54.6%

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

      8. *-rgt-identity 54.6%

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

   8. Applied egg-rr 54.6%

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

   if -5.00000000000000028e231 < (*.f64 z t) < 4.0000000000000002e206

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 63.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+231} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+239) (/ (/ (- x) t) z) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+239) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (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 ((z * t) <= (-1d+239)) then
        tmp = (-x / t) / z
    else
        tmp = x / (y - (z * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+239) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e+239:
		tmp = (-x / t) / z
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+239)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -1e+239)
		tmp = (-x / t) / z;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+239], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999991e238

   1. Initial program 74.6%

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

   3. Taylor expanded in y around inf 61.9%

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

      1. mul-1-neg 61.9%

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

      2. unsub-neg 61.9%

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

      3. associate-/l* 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in y around 0 74.6%

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

      1. associate-/r* 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. neg-mul-1 99.7%

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

   8. Simplified 99.7%

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

   if -9.99999999999999991e238 < (*.f64 z t)

   1. Initial program 98.2%

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

Alternative 8: 54.7% 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 95.4%

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

3. Taylor expanded in y around inf 53.3%

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

Developer Target 1: 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 2024150 
(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))))