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

Percentage Accurate: 95.7% → 95.7%

Time: 7.7s

Alternatives: 8

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: 95.7% 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: 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]

Derivation

1. Initial program 97.6%

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

3. Final simplification 97.6%

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

Alternative 2: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103} \lor \neg \left(z \leq 1.55 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.26e+103) (not (<= z 1.55e-154))) (/ (- x) (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.26e+103) || !(z <= 1.55e-154)) {
		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 <= (-1.26d+103)) .or. (.not. (z <= 1.55d-154))) 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 <= -1.26e+103) || !(z <= 1.55e-154)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.26e+103) or not (z <= 1.55e-154):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.26e+103) || !(z <= 1.55e-154))
		tmp = Float64(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 <= -1.26e+103) || ~((z <= 1.55e-154)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.26e+103], N[Not[LessEqual[z, 1.55e-154]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26000000000000006e103 or 1.54999999999999991e-154 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   5. Simplified 68.4%

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

   if -1.26000000000000006e103 < z < 1.54999999999999991e-154

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 74.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103} \lor \neg \left(z \leq 1.55 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.26e+103)
   (/ (/ x (- t)) z)
   (if (<= z 1.55e-154) (/ x y) (/ (- x) (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.26e+103) {
		tmp = (x / -t) / z;
	} else if (z <= 1.55e-154) {
		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 <= (-1.26d+103)) then
        tmp = (x / -t) / z
    else if (z <= 1.55d-154) 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 <= -1.26e+103) {
		tmp = (x / -t) / z;
	} else if (z <= 1.55e-154) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.26e+103:
		tmp = (x / -t) / z
	elif z <= 1.55e-154:
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.26e+103)
		tmp = Float64(Float64(x / Float64(-t)) / z);
	elseif (z <= 1.55e-154)
		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 (z <= -1.26e+103)
		tmp = (x / -t) / z;
	elseif (z <= 1.55e-154)
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.26e+103], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.55e-154], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26000000000000006e103

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 74.4%

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

      1. associate-*r/ 74.4%

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

      2. neg-mul-1 74.4%

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

   5. Simplified 74.4%

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

      1. neg-mul-1 74.4%

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

      2. *-commutative 74.4%

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

      3. times-frac 74.8%

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

   7. Applied egg-rr 74.8%

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

      1. associate-*l/ 74.9%

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

      2. associate-*r/ 74.9%

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

      3. associate-*l/ 74.8%

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

      4. frac-2neg 74.8%

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

      5. metadata-eval 74.8%

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

      6. associate-*l/ 74.9%

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

      7. *-un-lft-identity 74.9%

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

   9. Applied egg-rr 74.9%

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

   if -1.26000000000000006e103 < z < 1.54999999999999991e-154

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 74.8%

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

   if 1.54999999999999991e-154 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   5. Simplified 66.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.26e+103)
   (/ (/ (- x) z) t)
   (if (<= z 1.55e-154) (/ x y) (/ (- x) (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.26e+103) {
		tmp = (-x / z) / t;
	} else if (z <= 1.55e-154) {
		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 <= (-1.26d+103)) then
        tmp = (-x / z) / t
    else if (z <= 1.55d-154) 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 <= -1.26e+103) {
		tmp = (-x / z) / t;
	} else if (z <= 1.55e-154) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.26e+103:
		tmp = (-x / z) / t
	elif z <= 1.55e-154:
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.26e+103)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (z <= 1.55e-154)
		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 (z <= -1.26e+103)
		tmp = (-x / z) / t;
	elseif (z <= 1.55e-154)
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.26e+103], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.55e-154], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26000000000000006e103

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 74.4%

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

      1. associate-*r/ 74.4%

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

      2. neg-mul-1 74.4%

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

      3. *-commutative 74.4%

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

      4. associate-/r* 79.7%

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

   5. Simplified 79.7%

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

   if -1.26000000000000006e103 < z < 1.54999999999999991e-154

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 74.8%

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

   if 1.54999999999999991e-154 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   5. Simplified 66.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+144} \lor \neg \left(z \leq 1.55 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e+144) (not (<= z 1.55e-51))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+144) || !(z <= 1.55e-51)) {
		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 <= (-5.5d+144)) .or. (.not. (z <= 1.55d-51))) 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 <= -5.5e+144) || !(z <= 1.55e-51)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e+144) or not (z <= 1.55e-51):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e+144) || !(z <= 1.55e-51))
		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 <= -5.5e+144) || ~((z <= 1.55e-51)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e+144], N[Not[LessEqual[z, 1.55e-51]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000022e144 or 1.5499999999999999e-51 < z

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   5. Simplified 70.5%

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

      1. add-sqr-sqrt 34.7%

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

      2. sqrt-unprod 51.3%

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

      3. sqr-neg 51.3%

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

      4. sqrt-unprod 19.0%

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

      5. *-commutative 19.0%

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

      6. add-sqr-sqrt 38.8%

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

      7. *-un-lft-identity 38.8%

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

   7. Applied egg-rr 38.8%

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

      1. *-lft-identity 38.8%

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

      2. *-commutative 38.8%

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

   9. Simplified 38.8%

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

   if -5.50000000000000022e144 < z < 1.5499999999999999e-51

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf 68.9%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+144} \lor \neg \left(z \leq 1.55 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.6e-61)
   (/ x (* z t))
   (if (<= t 2.25e+133) (/ x y) (/ (/ x t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.6e-61) {
		tmp = x / (z * t);
	} else if (t <= 2.25e+133) {
		tmp = x / y;
	} else {
		tmp = (x / 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 (t <= (-3.6d-61)) then
        tmp = x / (z * t)
    else if (t <= 2.25d+133) then
        tmp = x / y
    else
        tmp = (x / t) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.6e-61) {
		tmp = x / (z * t);
	} else if (t <= 2.25e+133) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.6e-61:
		tmp = x / (z * t)
	elif t <= 2.25e+133:
		tmp = x / y
	else:
		tmp = (x / t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.6e-61)
		tmp = Float64(x / Float64(z * t));
	elseif (t <= 2.25e+133)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.6e-61)
		tmp = x / (z * t);
	elseif (t <= 2.25e+133)
		tmp = x / y;
	else
		tmp = (x / t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.6e-61], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.25e+133], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.60000000000000014e-61

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0 68.8%

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

      1. associate-*r/ 68.8%

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

      2. neg-mul-1 68.8%

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

   5. Simplified 68.8%

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

      1. add-sqr-sqrt 25.5%

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

      2. sqrt-unprod 46.3%

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

      3. sqr-neg 46.3%

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

      4. sqrt-unprod 24.2%

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

      5. *-commutative 24.2%

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

      6. add-sqr-sqrt 41.5%

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

      7. *-un-lft-identity 41.5%

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

   7. Applied egg-rr 41.5%

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

      1. *-lft-identity 41.5%

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

      2. *-commutative 41.5%

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

   9. Simplified 41.5%

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

   if -3.60000000000000014e-61 < t < 2.24999999999999992e133

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 70.2%

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

   if 2.24999999999999992e133 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in y around 0 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   5. Simplified 85.4%

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

      1. neg-mul-1 85.4%

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

      2. *-commutative 85.4%

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

      3. times-frac 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. associate-*l/ 89.0%

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

      2. associate-*r/ 89.0%

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

      3. associate-*l/ 88.9%

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

      4. frac-2neg 88.9%

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

      5. metadata-eval 88.9%

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

      6. associate-*l/ 89.0%

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

      7. *-un-lft-identity 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. add-log-exp 54.9%

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

       2. *-un-lft-identity 54.9%

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

       3. log-prod 54.9%

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

       4. metadata-eval 54.9%

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

       5. add-log-exp 89.0%

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

       6. associate-/l/ 85.4%

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

       7. *-commutative 85.4%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 51.9%

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

       10. sqr-neg 51.9%

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

       11. sqrt-unprod 45.6%

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

       12. add-sqr-sqrt 45.6%

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

   11. Applied egg-rr 45.6%

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

       1. +-lft-identity 45.6%

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

       2. associate-/r* 51.2%

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

   13. Simplified 51.2%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 1.0 (/ y x)))

C

double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

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 = 1.0d0 / (y / x)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

Python

def code(x, y, z, t):
	return 1.0 / (y / x)

Julia

function code(x, y, z, t)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 / (y / x);
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. clear-num 97.1%

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

   2. associate-/r/ 97.4%

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in y around inf 54.3%

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

   1. associate-*l/ 54.5%

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

   2. associate-/l* 54.5%

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

7. Applied egg-rr 54.5%

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

8. Final simplification 54.5%

   \[\leadsto \frac{1}{\frac{y}{x}} \]
9. 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 97.6%

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

3. Taylor expanded in y around inf 54.5%

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

4. Final simplification 54.5%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

Developer target: 96.4% 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 2024030 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

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

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