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

Percentage Accurate: 95.8% → 97.9%

Time: 6.0s

Alternatives: 6

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: 95.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+270) (/ (/ x (- t)) z) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -5e+270) {
		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) <= (-5d+270)) 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) <= -5e+270) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -5e+270:
		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) <= -5e+270)
		tmp = Float64(Float64(x / Float64(-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) <= -5e+270)
		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], -5e+270], 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) < -4.99999999999999976e270

   1. Initial program 70.2%

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

      1. clear-num 70.3%

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

      2. associate-/r/ 70.3%

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in y around 0 70.3%

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

      1. associate-*l/ 70.2%

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

      2. neg-mul-1 70.2%

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

      3. associate-/r* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -4.99999999999999976e270 < (*.f64 z t)

   1. Initial program 99.0%

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

Alternative 2: 76.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x) (* z t))))
   (if (<= (* z t) -4e+78)
     t_1
     (if (<= (* z t) 5e-69)
       (/ x y)
       (if (<= (* z t) 2e-43) t_1 (if (<= (* z t) 2e+25) (/ x y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double tmp;
	if ((z * t) <= -4e+78) {
		tmp = t_1;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = t_1;
	} else if ((z * t) <= 2e+25) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -x / (z * t)
    if ((z * t) <= (-4d+78)) then
        tmp = t_1
    else if ((z * t) <= 5d-69) then
        tmp = x / y
    else if ((z * t) <= 2d-43) then
        tmp = t_1
    else if ((z * t) <= 2d+25) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double tmp;
	if ((z * t) <= -4e+78) {
		tmp = t_1;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = t_1;
	} else if ((z * t) <= 2e+25) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -x / (z * t)
	tmp = 0
	if (z * t) <= -4e+78:
		tmp = t_1
	elif (z * t) <= 5e-69:
		tmp = x / y
	elif (z * t) <= 2e-43:
		tmp = t_1
	elif (z * t) <= 2e+25:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -4e+78)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-69)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e-43)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+25)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -x / (z * t);
	tmp = 0.0;
	if ((z * t) <= -4e+78)
		tmp = t_1;
	elseif ((z * t) <= 5e-69)
		tmp = x / y;
	elseif ((z * t) <= 2e-43)
		tmp = t_1;
	elseif ((z * t) <= 2e+25)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -4e+78], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-69], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-43], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+25], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.00000000000000003e78 or 5.00000000000000033e-69 < (*.f64 z t) < 2.00000000000000015e-43 or 2.00000000000000018e25 < (*.f64 z t)

   1. Initial program 92.9%

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

      1. clear-num 91.9%

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

      2. associate-/r/ 92.9%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. associate-*l/ 79.4%

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

      2. neg-mul-1 79.4%

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

      3. *-commutative 79.4%

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

   7. Applied egg-rr 79.4%

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

   if -4.00000000000000003e78 < (*.f64 z t) < 5.00000000000000033e-69 or 2.00000000000000015e-43 < (*.f64 z t) < 2.00000000000000018e25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 83.7%

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

Alternative 3: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{-t}}{z}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x (- t)) z)))
   (if (<= (* z t) -2e+94)
     t_1
     (if (<= (* z t) 5e-69)
       (/ x y)
       (if (<= (* z t) 2e-43)
         (/ (- x) (* z t))
         (if (<= (* z t) 2e+25) (/ x y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / -t) / z;
	double tmp;
	if ((z * t) <= -2e+94) {
		tmp = t_1;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+25) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / -t) / z
    if ((z * t) <= (-2d+94)) then
        tmp = t_1
    else if ((z * t) <= 5d-69) then
        tmp = x / y
    else if ((z * t) <= 2d-43) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d+25) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / -t) / z;
	double tmp;
	if ((z * t) <= -2e+94) {
		tmp = t_1;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+25) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / -t) / z
	tmp = 0
	if (z * t) <= -2e+94:
		tmp = t_1
	elif (z * t) <= 5e-69:
		tmp = x / y
	elif (z * t) <= 2e-43:
		tmp = -x / (z * t)
	elif (z * t) <= 2e+25:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(-t)) / z)
	tmp = 0.0
	if (Float64(z * t) <= -2e+94)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-69)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e-43)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e+25)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / -t) / z;
	tmp = 0.0;
	if ((z * t) <= -2e+94)
		tmp = t_1;
	elseif ((z * t) <= 5e-69)
		tmp = x / y;
	elseif ((z * t) <= 2e-43)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e+25)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+94], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-69], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-43], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+25], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e94 or 2.00000000000000018e25 < (*.f64 z t)

   1. Initial program 92.3%

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

      1. clear-num 91.2%

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

      2. associate-/r/ 92.3%

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. associate-*l/ 79.3%

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

      2. neg-mul-1 79.3%

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

      3. associate-/r* 82.5%

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

      4. distribute-neg-frac 82.5%

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

      5. distribute-neg-frac2 82.5%

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

   7. Applied egg-rr 82.5%

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

   if -2e94 < (*.f64 z t) < 5.00000000000000033e-69 or 2.00000000000000015e-43 < (*.f64 z t) < 2.00000000000000018e25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.6%

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

   if 5.00000000000000033e-69 < (*.f64 z t) < 2.00000000000000015e-43

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 86.0%

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

      1. associate-*l/ 86.0%

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

      2. neg-mul-1 86.0%

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

      3. *-commutative 86.0%

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

   7. Applied egg-rr 86.0%

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

Alternative 4: 78.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+59)
   (/ (/ x (- z)) t)
   (if (<= (* z t) 5e-69)
     (/ x y)
     (if (<= (* z t) 2e-43)
       (/ (- x) (* z t))
       (if (<= (* z t) 2e+25) (/ x y) (/ (/ x (- t)) z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+59) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+25) {
		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 ((z * t) <= (-2d+59)) then
        tmp = (x / -z) / t
    else if ((z * t) <= 5d-69) then
        tmp = x / y
    else if ((z * t) <= 2d-43) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d+25) 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 ((z * t) <= -2e+59) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 5e-69) {
		tmp = x / y;
	} else if ((z * t) <= 2e-43) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+25) {
		tmp = x / y;
	} else {
		tmp = (x / -t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+59:
		tmp = (x / -z) / t
	elif (z * t) <= 5e-69:
		tmp = x / y
	elif (z * t) <= 2e-43:
		tmp = -x / (z * t)
	elif (z * t) <= 2e+25:
		tmp = x / y
	else:
		tmp = (x / -t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+59)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 5e-69)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e-43)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e+25)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / Float64(-t)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e+59)
		tmp = (x / -z) / t;
	elseif ((z * t) <= 5e-69)
		tmp = x / y;
	elseif ((z * t) <= 2e-43)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e+25)
		tmp = x / y;
	else
		tmp = (x / -t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+59], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-69], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-43], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+25], N[(x / y), $MachinePrecision], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.99999999999999994e59

   1. Initial program 88.7%

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

      1. clear-num 87.6%

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

      2. associate-/r/ 88.7%

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in y around 0 75.8%

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

      1. associate-*l/ 75.8%

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

      2. neg-mul-1 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-/r* 85.2%

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

      5. distribute-neg-frac 85.2%

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

      6. distribute-neg-frac2 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -1.99999999999999994e59 < (*.f64 z t) < 5.00000000000000033e-69 or 2.00000000000000015e-43 < (*.f64 z t) < 2.00000000000000018e25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 84.6%

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

   if 5.00000000000000033e-69 < (*.f64 z t) < 2.00000000000000015e-43

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 86.0%

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

      1. associate-*l/ 86.0%

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

      2. neg-mul-1 86.0%

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

      3. *-commutative 86.0%

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

   7. Applied egg-rr 86.0%

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

   if 2.00000000000000018e25 < (*.f64 z t)

   1. Initial program 96.6%

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

      1. clear-num 95.7%

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

      2. associate-/r/ 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around 0 80.2%

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

      1. associate-*l/ 80.2%

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

      2. neg-mul-1 80.2%

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

      3. associate-/r* 77.1%

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

      4. distribute-neg-frac 77.1%

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

      5. distribute-neg-frac2 77.1%

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

   7. Applied egg-rr 77.1%

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

Alternative 5: 55.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+206}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x t) z)))
   (if (<= t -1.35e+29) t_1 (if (<= t 2.05e+206) (/ x y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / t) / z;
	double tmp;
	if (t <= -1.35e+29) {
		tmp = t_1;
	} else if (t <= 2.05e+206) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / t) / z
    if (t <= (-1.35d+29)) then
        tmp = t_1
    else if (t <= 2.05d+206) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / t) / z;
	double tmp;
	if (t <= -1.35e+29) {
		tmp = t_1;
	} else if (t <= 2.05e+206) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / t) / z
	tmp = 0
	if t <= -1.35e+29:
		tmp = t_1
	elif t <= 2.05e+206:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / t) / z)
	tmp = 0.0
	if (t <= -1.35e+29)
		tmp = t_1;
	elseif (t <= 2.05e+206)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / t) / z;
	tmp = 0.0;
	if (t <= -1.35e+29)
		tmp = t_1;
	elseif (t <= 2.05e+206)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t, -1.35e+29], t$95$1, If[LessEqual[t, 2.05e+206], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e29 or 2.0500000000000002e206 < t

   1. Initial program 90.4%

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

      1. clear-num 89.6%

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

      2. associate-/r/ 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in y around 0 66.1%

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

      1. associate-/r* 67.2%

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

      2. associate-*l/ 75.6%

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

   7. Applied egg-rr 75.6%

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

      1. associate-*l/ 75.7%

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

      2. neg-mul-1 75.7%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 41.7%

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

      5. sqr-neg 41.7%

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

      6. sqrt-unprod 18.3%

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

      7. add-sqr-sqrt 31.0%

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

   9. Applied egg-rr 31.0%

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

   if -1.35e29 < t < 2.0500000000000002e206

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf 63.1%

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

Alternative 6: 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 96.5%

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

3. Taylor expanded in y around inf 53.1%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. 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 2024052 -o generate:simplify
(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))))